lptv-aware bit loading and channel estimation in …lptv-aware bit loading and channel estimation in...

LPTV-Aware Bit Loading and Channel Estimation

in Broadband PLC for Smart Grid

By

Muharrem Ali Tunc

Submitted to the graduate degree program in the Department of Electrical Engineering& Computer Science and the Graduate Faculty of the University of Kansas

in partial fulfillment of the requirements for the degree ofDoctor of Philosophy.

Dissertation Committee:

Erik S. Perrins, Chairperson

Shannon D. Blunt

James P.G. Sterbenz

Lingjia Liu

Atanas Stefanov

Date Defended

The Dissertation Committee for Muharrem Ali Tunccertifies that this is the approved version of the following dissertation:

LPTV-Aware Bit Loading and Channel Estimationin Broadband PLC for Smart Grid

Erik S. Perrins, Chairperson

Date approved:

ii

Abstract

Power line communication (PLC) has received steady interest over recent decades because of

its economic use of existing power lines, and is one of the communication technologies envisaged

for Smart Grid (SG) infrastructure. However, power lines are not designed for data communica-

tion, and this brings unique challenges for data communication over power lines. In particular

for broadband (BB) PLC, the channel exhibits linear periodically time varying (LPTV) behav-

ior synchronous to the AC mains cycle. This is due to the time varying impedances of electrical

devices that are connected to the power grid. Another challenge is the impulsive noise in addi-

tion to power line background noise, which is due to switching events in the power line network.

In this work, we focus on two major aspects of an orthogonal frequency division multiplexing

(OFDM) system for BB PLC LPTV channels; bit and power allocation, and channel estimation

(CE).

First, we investigate the problem of optimal bit and power allocation, in order to increase

bit rates and improve energy efficiency. We present that the application of a power constraint

that is averaged over many microslots can be exploited for further performance improvements

through bit loading. Due to the matroid structure of the optimization problem, greedy-type

algorithms are proven to be optimal for the new LPTV-aware bit and power loading. Significant

gains are attained especially for poor (i.e. high attenuation) channel conditions, and at reduced

transmit-power levels, where the energy per bit-transmission is also low. Next, two mechanisms

are utilized to reduce the complexity of the optimal LPTV-aware bit loading and peak microslot

power levels: (i) employing representative values from microslot transfer functions, and (ii)

power clipping. The ideas of LPTV-aware bit loading, complexity reduction mechanism, and

iii

power clipping are also applicable to non-optimal bit loading schemes. We apply these ideas to

two additional sub-optimal bit loading algorithms that are based on even-like power distribution

for a portion of the available spectrum, and demonstrate that similar gains in bit rates are

achieved.

Second, we tackle the problem of CE for BB PLC LPTV channels. We first investigate

pilot based CE with different pilot geometry in order to reduce interpolation error. Block-type,

comb-type, and incline type pilot arrangements are considered and a performance comparison

has been made. Next we develop a robust CE scheme with low overhead that addresses the

drawbacks of block-type pilot arrangement and decision directed CE schemes such as large

estimation overhead for block-type pilot geometry, and difficulty in channel tracking in the case

of sudden changes in the channel for decision directed approaches. In order to overcome these

drawbacks, we develop a transform domain (TD) analysis approach to determine the cause

of changes in the channel estimates, which are due to changes in the channel response or the

presence of impulsive noise. We then propose a robust CE scheme with low estimation overhead,

which utilizes pilot symbols placed widely apart and exploits the information obtained from TD

analysis as a basis for switching between various CE schemes. The overhead of the proposed

scheme for CE is low, and sudden changes in the channel are tracked affectively. Therefore, the

effects of the LPTV channel and the impulsive noise on CE are mitigated.

Our results indicate that for bit and power allocation, the proposed reduced complexity

LPTV-aware bit loading with power clipping algorithm performs very close to the optimal

LPTV-aware bit loading, and is an attractive solution to bit loading in a practical setting.

Finally, for the CE problem, the proposed CE scheme based on TD analysis has low estimation

overhead, performs well compared to block-type pilot arrangement and decision directed CE

schemes, and is robust to changes in the channel and the presence of impulsive noise. Therefore,

it is a good alternative for CE in BB PLC.

iv

Acknowledgments

Praise be to God, the Most Gracious and the Most Merciful...

First and foremost, I would like to thank Dr. Erik Perrins for his excellent guidance and

support during my studies. Without his support and guidance, this work would not have been

accomplished. I would also like to thank Dr. Lutz Lampe for his guidance and collaboration.

I would like to thank Dr. James Sterbenz and Dr. Shannon Blunt, from whom I took classes

and learned a lot.

I also thank my family: my wife Sevde, my mother and father to whom I dedicate this

dissertation, and my brothers Gokturk and Yavuz for their support throughout my studies.

Special thanks to my friends Fatih, Zeynep, and Oguzhan who have always been much more

than a friend to me. Many thanks to Mustafa, Esin, Erkan, Hale, Ferhat, Rabia, Neslihan,

Kursat, Selahaddin, Abdulbaki, Ismet, Murat, Erhan, Eyyup, Akif, Mucahit, Oguz, Songul,

Elmeddin, Nurullah, Senem, Pasha, Anastasia, Cal, Jolene, and Steve (and many others that

I could not list here), for their invaluable friendship. Finally, thanks to my classmates Ehsan,

Zaid, Daniel, Egemen, and Cenk for their valuable company and friendship...

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vi

Table of Contents

Abstract iii

Table of Contents vii

List of Figures xi

1 Introduction and Motivation 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Power Line Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 BB PLC Channel Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Outline of the Chapters that Follow . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 System Model 9

2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Channel Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 OFDM System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 DFT Implementation of an OFDM System . . . . . . . . . . . . . . . . . 16

3 Bit and Power Allocation 19

3.1 Key Points of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Channel Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Greedy Algorithm for Optimal Bit Allocation . . . . . . . . . . . . . . . . 25

3.4.2 Simplistic Adaptation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Optimal LPTV-aware Bit Loading Scheme . . . . . . . . . . . . . . . . . . . . . . 30

3.5.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii

3.5.2 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.3 Resultant Power Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5.4 Energy Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.5 Power Saving in BB PLC Standards . . . . . . . . . . . . . . . . . . . . . 37

3.6 Sub-optimal Reduced Complexity LPTV-aware Bit Loading . . . . . . . . . . . . 39

3.6.1 Reduced Complexity LPTV-aware Bit Loading . . . . . . . . . . . . . . . 39

3.6.2 Power Clipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6.3 Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6.4 Peak-to-average Power Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7 LPTV-aware Bit Loading for Non-optimal Algorithms . . . . . . . . . . . . . . . 43

3.7.1 Algorithm A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7.2 Algorithm B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Channel Estimation 53

4.1 Key Points of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.1 LS and LMMSE Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.2 Pilot-based Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.3 Decision-directed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Transform Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Performance Analysis and Proposed Final Scheme . . . . . . . . . . . . . . . . . 69

4.5.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5.2 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.3 Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5.4 Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5.5 Proposed Final Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5.6 Optimal Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Conclusion 81

5.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Areas of Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

viii

A Matroid Structures 85

A.1 Matroids and Greedy Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.2 Integer Bit Loading as a Matroid Structure . . . . . . . . . . . . . . . . . . . . . 87

B Channel Interpolation 91

B.1 Estimation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References 95

ix


x

List of Figures

1.1 Development of optimal and reduced complexity LPTV-aware bit loading schemes. 7

2.1 LPTV channels: (a) commuted; (b) harmonic. . . . . . . . . . . . . . . . . . . . . 10

2.2 Network topology used for the channel model [11, Fig. 1] . . . . . . . . . . . . . . 11

2.3 Illustration of division of available bandwidth in an OFDM system. . . . . . . . . 13

2.4 Illustration of an OFDM system for PLC. . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Increase in channel capacity due to channel adaptation for commuted and har-

monic channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Increase in raw bit rate due to the simplistic adaptation scheme compared to the

non-adaptation scheme for commuted and harmonic channels. . . . . . . . . . . . 30

3.3 Raw bit rate for the non-adaptation and simplistic adaptation schemes for com-

muted channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Increase in raw bit rate due to the optimal LPTV-aware bit loading compared

to the simplistic adaptation scheme for commuted and harmonic channels. . . . . 33

3.5 Raw bit rate for simplistic adaptation and optimal LPTV-aware bit loading

schemes for commuted channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Resultant microslot transmit signal PSD levels for Channel 5, with -75 dBm/Hz

average transmit signal PSD over one AC mains cycle, for the optimal LPTV-

aware bit loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7 Energy per one bit transmission for the optimal LPTV-aware bit loading scheme

in commuted and harmonic channels. . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8 Raw bit rate for the sub-optimal reduced complexity LPTV-aware bit loading

with power clipping, and the optimal LPTV-aware bit loading schemes in com-

muted and harmonic channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.9 Algorithm A LPTV-aware, Algorithm B LPTV-aware, and Algorithm B reduced

complexity LPTV-aware bit loading with power clipping using the maximum

magnitude values for Hj for commuted channels. . . . . . . . . . . . . . . . . . . 46

xi

3.10 Algorithm A LPTV-aware, Algorithm B LPTV-aware, and Algorithm B reduced

complexity LPTV-aware bit loading with power clipping using the maximum

magnitude values for Hj for harmonic channels. . . . . . . . . . . . . . . . . . . . 48

3.11 Simplistic channel adaptation with Algorithm B, and Algorithm B LPTV-aware

bit loading for commuted channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.12 Simplistic channel adaptation with Algorithm B, and Algorithm B LPTV-aware

bit loading for harmonic channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Normalized mean square error for Rhh computation using different number of

distinct transfer functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Different types of pilot arrangement: (a) block-type pilots; (b) comb-type pilots;

(c) comb-type pilots where pilots are placed widely apart; (d) incline-type pilots 59

4.3 Normalized mean square error using linear interpolation, L = 5. . . . . . . . . . 60

4.4 Normalized mean square error using linear interpolation, L = 10. . . . . . . . . . 61

4.5 Normalized mean square error using cubic interpolation, L = 5. . . . . . . . . . . 63

4.6 Normalized mean square error using cubic interpolation, L = 10. . . . . . . . . . 65

4.7 TD analysis for a single realization of four cases for LS estimates . . . . . . . . . 66

4.8 Low frequency metric for Cases A–D. . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.9 High frequency metric for Cases A–D. . . . . . . . . . . . . . . . . . . . . . . . . 68

4.10 Case A channel estimates for the various schemes. The amplitude and phase of

the channel transfer function are illustrated. . . . . . . . . . . . . . . . . . . . . . 72

4.11 Case B channel estimates for the various schemes. The amplitude and phase of


4.12 Case C channel estimates for the various schemes. The amplitude and phase of


4.13 Case D channel estimates for the various schemes. The amplitude and phase of


4.14 Proposed final channel estimation scheme based on TD analysis. . . . . . . . . . 78

A.1 Strictly increasing concave rate function illustrated as a function of overall SNR

with ideal case of no SNR-gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.1 Linear and cubic interpolation kernels. . . . . . . . . . . . . . . . . . . . . . . . . 92

B.2 Estimation error decomposition [22, Fig. 3] . . . . . . . . . . . . . . . . . . . . . 93

xii

Chapter 1

Introduction and Motivation

1.1 Introduction

In this effort, two major aspects of an orthogonal frequency division multiplexing (OFDM)

communication system for broadband (BB) power line communication (PLC) are addressed; (i)

bit and power allocation, and (ii) channel estimation (CE). Optimal and reduced complexity

bit and power allocation algorithms, and a robust CE scheme with low estimation overhead are

developed.

1.2 Power Line Communication

PLC [1] has been a topic of continued interest over time because it allows for dual use of

existing power line infrastructure. This economic advantage makes PLC a focus of study for

notable applications that include broadband distribution over power lines (BPL), automatic

metering infrastructure (AMI), and Smart Grid (SG) communications infrastructure, to name

a few [2].

Historically, early in the twentieth century, PLC was used for switching in substations,

metering and basic load control. The interest for PLC research gained momentum in 1980s

and increased further in 1990s. Much research has been done since then, along with products

developed in the industry during this time. Power lines are usually classified into high (>100

1

kV), medium (1–100 kV) and low (<1 kV) voltage networks, with increasing challange for

communication respectively. LV distribution networks have been the focus of the most research

in PLC, which provides easy access to the network in various buildings. The utility companies

have had interest in PLC with the motovation of load management, which requires switching on

and off devices at the times of peak power demand. Another incentive for electric companies has

been automatic metering to reduce costs in labor, and utilization of PLC in SG infrastructure.

In the case of power outages in the cellular and wireless communication networks, PLC can also

play an important role in disaster recovery for communication over some critical links, where

there may be no other technology available for communication [1].

One challenge for the development of PLC technology has been the widely different wiring

and grounding practices in different parts of the world such as Europe and United States.

International harmonization efforts have been initiated in the past 20–30 years, and many

regulatory bodies (for instance, US National Electric Code (NEC)) have mandated certain set

of practices for wiring. Although there is still no set of practices that is followed all around the

world, these regulations helped greatly simplify the analysis of signal transmission over power

lines [1].

The PLC technology can be grouped into three categories based on the bandwidth they

occupy: 1) The Ultra narrow band (UNB) technology provides very low data rates in the order

of hundred bits per second (bps) in the ultra low 0.3–3 kHz frequency band. Some applications

of UNB technology are the Ripple Carrier Signaling that supports load control applications,

AMR turtle system, and Two-Way Automatic Communications System (TWACS). 2) The

Narrowband (NB) technology operates in 3–148.5 kHz CENELEC (European Committee for

Electrotechnical Standardization) frequency band in Europe, and 10–490 kHz FCC frequency

band in United States. Within this category, single carrier low data rate (LDR) solutions can

provide data rates of a few kbps, and multicarrier high data rate (HDR) solutions can provide

up to 500 kbps. 3) The Broadband (BB) technology operates in 1.8–250 Mhz frequency band,

can provide up to a few hundred Mbps data rate, and is the focus of our work [2].

2

Some of the applications of BB PLC include the collection and distribution of data on energy

consumption of smart appliances, demand response and management programs, dynamic pric-

ing and flexible power control of appliances, communication between Plug-in Electric Vehicles

(PEV) and their charging stations, and communications between smart appliances. These ap-

plications are envisaged for the in-home (IH) domain of SG, and often require devices with low

complexity and reduced power consumption, since they will likely be deployed in large quantities

in the SG [3].1 These criteria are essential for technologies targeting the IH domain of SG [4].

The following BB PLC standards are worth mentioning in this domain: IEEE 1901 [5], ITU-T

G.hn [3], and the HomePlug Green PHY (HPGP) [4] specification. Each of these standards has

design features that are relevant to our work in terms of reducing the power consumption and

providing low complexity solutions.

1.3 BB PLC Channel Characteristics

Two important characteristics of the BB power line channel are; (i) its transfer function

exhibits a linear periodically time varying (LPTV) behavior [6], and (ii) impulsive noise is

present in addition to the power line background noise. The LPTV behavior of the BB PLC

channel stems from the time-varying impedances of the electrical devices that are connected

to the power grid and the impulsive noise is caused by the switching events in the power line

network. These channel and noise characteristics must be taken into account when designing a

communication system for BB PLC.

1.4 Problem Statement

In this work, we address two aspects of an OFDM system for BB PLC LPTV channels: i)

bit and power allocation, and ii) CE.

1There is also a valid discussion about the application of BB versus narrowband PLC technology for SG, con-sidering that envisaged use cases often require moderate data rates and device complexity and power consumptionare reduced for narrowband systems. But these considerations are outside the scope of this work focusing on BBPLC.

3

For bit and power allocation, conventional algorithms designed for LPTV channels in the

literature adapt to the LPTV channel via so called microslots in time. However, these algorithms

do not consider the time-varying conditions of the channel transfer function over one AC mains

cycle and allocate power to each microslot regardless of its channel condition (i.e. channel

attenuation). Therefore, these algorithms are not optimal in terms of maximizing the achieved

data rates. In this work, we show that the optimal bit and power allocation algorithm for

LPTV channels can be developed by considering the fluctuations in the transfer function over

one AC mains cycle and allocating power accordingly. This results in significant improvement

in achieved data rates. Next, we show that the complexity of the optimal algorithm can be

reduced if the task of bit and power allocation over one AC mains cycle is divided into smaller

tasks, which makes the algorithm work on a smaller data set.

For the CE aspect, the LPTV channel requires a large number of transfer functions to be

estimated over one AC mains cycle. If we assume that the channel remains constant for one

OFDM symbol time, and can vary from one OFDM symbol to the next, depending on the

channel bandwidth this may result in hundreds of microslots for CE during one AC mains cycle

(i.e. 20 ms. for 50 Hz. in Europe and 16.6 ms. for 60 Hz. in United States). When pilot

symbols are utilized with high density—for instance in the case of block-type pilot geometry—

to estimate the channel, the estimation overhead is large. When decision directed approaches

are utilized, the overhead is minimal, but the channel tracking capabilities are limited in the

case of abrupt changes in the channel transfer function and noise due to the LPTV channel and

the impulsive noise. Therefore, a low overhead CE scheme that is robust to abrupt changes in

the channel transfer function and considers the presence of impulsive noise is needed. In this

work, we first examine pilot based CE with various pilot geometry to reduce interpolation error

inherent in comb-type pilot arrangement. We then develop a transform domain (TD) analysis

that examines the change in two consecutive channel estimates in a TD, and determines the

changes in the estimate due to a change in the transfer function or the presence of impulsive

noise. Next, a final CE scheme that switches between various CE schemes based on the result

4

of TD analysis for different cases is developed. The proposed scheme has low overhead, and is

robust to changes in the transfer function and the noise.

1.5 Contributions

For the bit loading aspect of an OFDM system, we consider two essential design criteria for

BB PLC devices for the IH domain of SG; (i) reducing the power consumption (ii) reducing the

complexity. We address these two criteria in two steps. In the first step, we address the problem

of optimal bit and power allocation for BB PLC LPTV channels, which increases the achieved

data rate, and in return reduces the power consumption. In the second step, we address the

issue of complexity, and provide mechanisms that reduce the complexity of the proposed bit

loading scheme, while maintaining a near-optimal performance.

• Step 1: It was shown in the previous works that some improvement in bit loading can

be achieved simply by adapting to the LPTV channel for a BB PLC OFDM system via

microslots, on which bit loading is based on. Our solution to the problem of bit loading

also uses microslots, except it makes the bit loading procedure fully aware of the LPTV

behavior of the channel in order to achieve higher data rates, and consequently significant

power savings. In this step, we carefully formulate the optimal integer bit loading problem

for LPTV channels as a matroid structure and show that greedy-type algorithms yield

the maximum data rate when combined with our LPTV fully aware bit loading scheme.

• Step 2: As for the second step, our solution involves reducing the complexity of the

system by making use of representative values from microslot transfer functions, instead

of using a long combined transfer function. Our approach considers the problem of bit and

power allocation during one AC mains cycle as a series of two simpler tasks: i) finding out

the power to allocate to microslots within one AC mains cycle using representative values,

ii) finding out bit and power allocation for subchannels of a microslot using the power

levels computed in the initial step. Our proposed schemes result in a fluctuation of the

5

microslot power levels within one AC cycle of the mains, rather than a constant power level

for all microslots. In order to avoid peak power levels, the reduced complexity bit loading

scheme enables a power clipping algorithm that limits the total power to allocate to a

microslot. Additionally, we analyze the energy efficiency, microslot power level variations,

and peak-to-average power ratio (PAPR) analysis for the proposed bit loading schemes.

We observe that transmitting at reduced power levels can be more energy efficient, where

we also have bigger gain due to the proposed bit loading schemes, and power clipping is

not necessary.

Finally, LPTV-aware bit loading, reducing complexity, and power clipping can also be applied

to non-optimal bit loading algorithms, since these ideas are independent of the underlying

bit loading algorithm. In order to quantify the improvement for non-optimal algorithms, in

addition to the optimal greedy-type algorithm, these ideas are implemented for two bit loading

algorithms that are based on even-like power distribution. Our results indicate that similar

gains in bit rates are achieved.

For the CE aspect, the existing CE schemes such as pilot-based and decision directed ap-

proaches have certain drawbacks. Pilot based schemes may result in high estimation overhead

when pilot symbols are placed densely in the OFDM symbol (i.e. block-type pilot arrangement,

all subchannels carry pilot symbols.), since the number of channels to estimate during one AC

mains cycle in large. On the other hand, decision directed approaches have low estimation

overhead but their channel tracking capabilities in the case of abrupt changes in the channel

transfer function and presence of impulsive noise is low. First, we analyze least squares (LS) and

linear minimum mean squared error (LMMSE) algorithms for CE from certain practical consid-

erations. Next, we examine pilot based CE with various pilot geometry; block-type, comb-type,

and incline-type. Incline-type pilot arrangement is considered to help reduce interpolation error

inherent in comb-type pilot arrangement, and a comparison is made.

We then develop the TD analysis approach. By comparing the estimates for two consecutive

OFDM symbols in a TD, the analysis determines if the channel has changed or if the impulsive

6

noise is present. High and low frequency metrics in TD are defined to quantify the effect of

changes in the channel response and noise. TD analysis provides useful information for CE

even when pilots are spaced widely apart, which results in low estimation overhead. Next, we

develop a robust CE scheme that is based on TD analysis, has low overhead, and is robust

to changes in the LPTV channel and noise. The proposed scheme has the ability to switch

between various CE schemes based on the result of TD analysis, unlike conventioanal schemes

that are based on a single scheme in all cases.

Developed here for the first time

BB LPTV Channelmeasurements and

modeling, [Corripio et al.

2006, 2011].

Simplistic adaptationfor OFDM bit loading[Katar et al. 2006, Honda et al. 2008,Kim et al. 2009].

Chapter 3Optimal / Reduced

complexityLPTV-aware bit loading,

Schemes #3-4.

Chapter 4Incline-type pilots

implemented for BB PLC,TD analysis, Robust CE

with low estimationoverhead.

Pilot based channel estimation for PLC

[Cortes et al. 2007, Tonelloet al. 2009],

Transform domain processing

[Zhao et al. 1997].

Figure 1.1. Development of optimal and reduced complexity LPTV-aware bitloading schemes.

In summary, our contributions in this work are depicted in Figure 1.1. As for the bit loading

in our work, we refer to four bit loading schemes that are of interest: #1 the non-adaptation

scheme, where the lowest transfer function is utilized for all microslots [7]; #2 the simplistic

adaptation scheme that is not fully LPTV-aware, where distinct transfer functions are exploited

for each microslot but the microslots allocations are independent of each other [8, 9, 7]; #3 the

optimal LPTV-aware scheme (as pointed out above), and #4 the sub-optimal (practical) LPTV-

7

aware scheme, which has notably lower complexity and incorporates power clipping. Schemes

#3 and #4 are central contributions of this work for the bit loading aspect.

As for the CE aspect, our contribution for pilot-based CE is the application of incline-type

pilot arrangement for BB PLC channels to help reduce interpolation error. Additionally, TD

analysis approach and the robust CE scheme with low overhead based on TD analysis are our

main contributions for CE.

1.6 Outline of the Chapters that Follow

In Chapter 2, the channel model used to generate LPTV channels throughout this work

and OFDM system are explained. The channel capacity computation for non-adaptation and

adaptation cases and the formulation of the optimal bit loading algorithm based on a greedy-

type algorithm is developed in Chapter 3. Next, the mechanisms that reduce the complexity

of the optimal bit loading algorithm and peak power levels are explained. The performance

evaluation for even-like power distribution algorithms is in Section 3.7.

In Chapter 4, various pilot arrangement for pilot-based CE are examined, and TD analysis

based on pilots placed widely apart is developed. Then, a robust CE scheme with low estimation

overhead is designed for LPTV channels that exploits the result of TD analysis, and is compared

to pilot-based and decision-directed CE schemes. Finally, we offer conclusions in Chapter 5.

8

Chapter 2

System Model

In this chapter, we explain the channel model used to generate LPTV channels for analysis

and OFDM system utilized in this dissertation.

2.1 Channel Model

The LPTV behavior of the BB PLC channel stems from the time-varying impedances of

the electrical devices that are plugged into the power grid and it manifests itself in two distinct

forms. Some devices (such as low power lamps and light dimmers) have two separate impedances

during one AC mains cycle and these induce a commuted LPTV channel, which alternates

sharply between high and low values. Other devices (such as monitors and microwave ovens)

have smooth time variations in their impedances and these lead to a harmonic LPTV channel,

which is a combination of several transfer functions that have a progressive variation within

one cycle of the mains [6, 10]. Examples of commuted and harmonic channels are illustrated in

Fig. 2.1.

In this work, we adopt the channel model in [11] to generate BB LPTV channels. The model

generates realistic LPTV channel realizations based on a simplified parametric representation

of a PLC network as shown in Fig. 2.2 from [11, Fig. 1]. The transmitter and the receiver are

connected via four line segments L1. . .L4, and three loads are connected to the power grid with

9

Figure 2.1. LPTV channels: (a) commuted; (b) harmonic.

distances S1. . .S3. The lengths of the line segments and the distances of the loads are adjusted

in the model to generate channels that correspond to different topologies.

The loads are modeled to be selective in frequency and time. Three types of impedances

are considered:

1. Approximately constant impedances

2. Time-invariant, but frequency-selective

3. Time-varying and frequency selective.

The constant impedances can be {5, 50, 150, 1000,∞}, which stand for low, RF standard, similar

to transmission line Z0, high, and open circuit impedances, respectively. Time-invariant and

10

L1 L2 L3 L4

S1 S2 S3

ZG

Z1 Z2 Z3

ZL

Figure 2.2. Network topology used for the channel model [11, Fig. 1]

frequency-selective impedances can be described by

Z(w) =R

1 + jQ(ww0− w0

w

) , (2.1)

where R is the resistance at resonance, w0 is the resonance angular frequency, and Q is the qual-

ity factor to determine selectivity. Finally, time varying impedances can have either commuted

behavior between two values ZA and ZB synchronous to the AC mains cycle, or a harmonic

behavior

Z(w, t) = ZA(w) + ZB(w)| sin(2π

T0t+ φ)|, 0 ≤ t ≤ T0, (2.2)

where ZA is the offset impedance, ZB reflects the amplitude of variation, T0 is the duration

of one AC mains cycle, and φ is the variation with respect to the mains voltage zero-crossing

point.

In both commuted and harmonic channels, the AC mains cycle is divided into M microslots

in time, where each microslot has a distinct transfer function, Hj(f), j ∈ {0, . . . ,M − 1}. The

model in [11] uses M = 50, and in each microslot the spectrum is divided into N subchannels,

for example, later we choose N = 256. Hj,i is then the value of the channel transfer function

Hj(f) in the i-th subchannel, i ∈ {0, . . . , N − 1}.

11

2.1.1 Channel Parameters

In order to have some measure of the strength of the LPTV behavior for an LPTV channel,

we define the excursion parameter for the i-th subchannel in the j-th microslot as

γj,i =|Hj,i||Hlow,i|

, (2.3)

where Hlow,i is the lowest of all transfer functions in the i-th subchannel and is defined as

Hlow,i = Hjlow,i : jlow = argmin0≤j≤M−1

|Hj,i|. (2.4)

A related quantity, Hhigh,i, is defined implementing argmax instead of argmin in (2.4). We then

define [γ]avg as the average of γj,i values—in dB—over all subchannels and microslots:

[γ]avg =1

MN

M−1∑j=0

N−1∑i=0

20 log10(γj,i), (2.5)

which is an overall indicator of how far apart the microslot transfer functions Hj(f) are from

Hlow,i. The rationale behind averaging γj,i values in dB, rather than averaging in linear scale

and then converting to dB, will be justified in Section 3.4 along with our channel capacity

calculations. As for the channel attenuation, we define the average attenuation for an LPTV

channel as

[Havg] = 10 log10

1

MN

M−1∑j=0

N−1∑i=0

|Hj,i|2 . (2.6)

These parameters defined in this section help characterize LPTV channels in the following

chapters based on the channel conditions (i.e. low, medium, high channel attenuation) and

LPTV channel strength.

12

2.2 OFDM System

OFDM [12] is a multicarrier modulation that divides the available bandwidth W into many

subchannels with relatively narrower width ∆f = W/N as illustrated in Fig. 2.3, where N

is the number of subchannels. The signal in each subcarrier can be independently coded and

modulated with the rate 1/∆f , via the bit loading process. A small enough ∆f results in a

constant channel response at each subchannel. The carrier sinusoid for the subchannel i is

si(t) = cos(2πfit), i = 0, 1, 2, . . . , N − 1, (2.7)

where fi is the mid frequency in the i-th subchannel. The symbol rate 1/T is chosen to be

|H(f)|

Wf

0

Figure 2.3. Illustration of division of available bandwidth in an OFDM system.

equal to ∆f , such that the integral

∫ T

0cos(2πfit+ φi) cos(2πflt+ φl)dt = 0, (2.8)

is zero for one symbol time, where fi−fl = n∆f, n = 1, 2, . . . , N−1. Therefore, the subchannels

are mutually orthogonal.

13

For an OFDM system, the symbol rate is N times less than a single carrier system that

occupies the whole bandwidth W . Thus, the arrival time between two symbols is T = NTs,

where Ts = 1/W is the symbols time for the single carrier system. When N is large, symbol

arrival time can become much larger than the channel-time dispersion, which helps reduce

intersymbol interference. The M -ary quadrature amplitude modulation (QAM) time domain

signal on the i-th subchannel is

ui(t) =

√2

TAic cos(2πfit)−

√2

TAis sin(2πfit)

= Re

[√2

TAie

jθiej2πfit

]

= Re

[√2

TXie

j2πfit

], (2.9)

where

Xi = Aiejθi (2.10)

is the complex signal from available M -ary QAM constellation on the i-th subcarrier, and

Ai =√A2ic +A2

is (2.11)

θi = tan−1(AisAic

). (2.12)

For large N , the channel frequency response can be approximated to a constant

Hi(fi) = Hi = |Hi|ejφi , (2.13)

where the received symbol at subchannel i in time domain is

ri(t) =

√2

T|Hi|Aic cos(2πfit+ φi) +

√2

T|Hi|Ais sin(2πfit+ φi) + ni(t)

= Re

[√2

THiXie

j2πfit + ni(t)

], (2.14)

14

where ni(t) is the additive noise in the i-th subchannel. At the receiver, assuming CE is done a

priori, the received signal can be demodulated by cross-correlating the received symbol at i-th

subchannel with two basis functions

ψ1(t) =

√2

Tcos(2πfit+ φi), 0 ≤ t ≤ T

ψ2(t) = −√

2

Tsin(2πfit+ φi), 0 ≤ t ≤ T, (2.15)

and sampling at T . The received signal vector is then

yi = (|Hi|Aic + ηir, |Hi|Ais + ηii), (2.16)

where

ηi = ηir + jηii (2.17)

is the noise expressed as a complex number. yi can also be expressed as a complex number

Yi = HiXi + ηi. (2.18)

The channel effect can be undone by

Y ′i =YiHi

= Xi + η′i, (2.19)

where

η′i =ηiHi. (2.20)

The detector then can compute the distance between Yi and possible QAM constellation points,

and selects the signal with the shortest distance. The resultant OFDM system with the im-

plementation of two cross-correlators for each subchannel requires 2N cross-correlators, or 2N

matched filters at the demodulator, and 2N parallel filters at the modulator. The system can

rather be implemented by the equivalent discrete Fourier transform (DFT) and its inverse, or

15

Information bits

mapping,S/P

.

.

.

.

.

Hermitiansymmetry,modulator(IDFT)

cyclic prefix added, P/S

.

.

.

.

.

D/A converter

bits parallel toserialconversion

.

.

.

.

.

demodulator(DFT)

cyclic prefix removal,

serial to

parallelconversion

.

.

.

.

.

A/D converter

Power line channel

powerlinenoise

channel estimation,equalization,detection

.

.

.

.

.

Figure 2.4. Illustration of an OFDM system for PLC.

its efficient implementation fast Fourier transform (FFT) and the inverse [12].

2.2.1 DFT Implementation of an OFDM System

The multicarrier OFDM system implemented in this work is illustrated in Fig. 2.4. Since,

the available bandwidth for communication is divided into many subchannels, the equalization is

a simple task at the receiver. At the transmitter, the information bits are grouped and mapped

into symbols from available complex constellations at each subcarrier i, Xi. Then, the OFDM

signal can be obtained by an IDFT. If N is the number of subcarriers, N complex frequency

domain symbols Xi can be arranged into 2N Hermitian symmetric symbols. The output of

the 2N -point IDFT is then 2N real samples of the time domain signal. A cyclic prefix (CP)

is added to the OFDM symbol in order to eliminate inter-symbol interference (ISI) due to the

multipath channel. The CP must be longer than the channel delay, and is copied from samples

at the end of the OFDM symbol. The OFDM signal then goes through the power line channel

and power line noise is added to the signal. At the receiver, the operation at the transmitter is

reversed. First, the CP is removed, then a 2N -point DFT is performed on the received symbol.

Due to the Hermitian symmetry of the DFT operation on a real sequence, only N complex

symbols out of 2N are passed on to the next stage for CE. The received demodulated noisy

16

OFDM symbol y can be described in the frequency domain as:

y = Xh + n, (2.21)

where X is an N×N matrix with all zeros except the diagonal terms being transmitted symbols

Xi, i = 0 . . . N − 1, h is the N × 1 channel frequency response, and n is the N × 1 complex

additive noise.

In this work, we use N = 256, and QAM is implemented in each subcarrier. The channel

h[n] is implemented by NB = 50 real samples of a finite impulse response (FIR) filter, and

the CP is the last NB samples of the OFDM symbol. The transmitted signal’s power spectral

density is -55 dBm/Hz, and we assume that the channel transfer function remains the same

for the duration of one OFDM symbol. The power line background noise is additive white

Gaussian noise (AWGN), with N (0, σ2) -110 dBm/Hz PSD, and the impulsive noise is modeled

as N (0,Kσ2), where K = 100 [13].

17


18

Chapter 3

Bit and Power Allocation

3.1 Key Points of the Chapter

• Previous work has demonstrated that improvements in bit loading can be achieved with

LPTV channel adaptation via microslots in time for an OFDM system.

• The application of a power constraint that is averaged over many microslots can be ex-

ploited for further performance improvements through loading.

• Due to the matroid structure of the optimization problem, greedy-type algorithms are

proven to be optimal for the new LPTV-aware bit and power loading.

• Significant gains are attained especially at reduced transmit-power levels, where the energy

per bit-transmission is also low, and for poor (i.e. high attenuation) channel conditions.

• Two mechanisms are utilized to reduce the complexity of the optimal LPTV-aware bit

loading and peak microslot power levels: (i) employing representative values from mi-

croslot transfer functions, and (ii) power clipping.

• The reduced complexity LPTV-aware bit loading with power clipping performs very close

to the optimal LPTV-aware bit loading, which makes it an attractive option in a practical

setting.

19

• The idea of LPTV-aware bit loading, complexity reduction via representative values, and

power clipping are also applicable to non-optimal bit loading algorithms, for which similar

increase in bit rates are observed.

3.2 Introduction

Two crucial characteristics of the BB power line channel are that its transfer function

exhibits a linear periodically time varying (LPTV) behavior [6], and impulsive noise is present

in addition to the power line background noise due to switching events; the former of these two

is our focus in this chapter.

The BB power line channel characteristics have been the subject of recent study. The

authors in [14] derive an algorithm to detect the transition between transfer functions of the

LPTV channel to overcome the effects of periodic switching events. In [7], an adaptation

scheme is presented that allows for the BB PLC cyclo-stationary noise and results in higher

data rates. In [15], the effect of the LPTV channel on the performance of several commercial

PLC adapters is studied. In [8], a fast bit loading algorithm for orthogonal frequency division

multiplexing (OFDM) is proposed that adapts to the BB LPTV channel; similarly, in [9] a

scheme is developed that adapts to the LPTV channel and the noise synchronized with the AC

mains cycle. In [16], the optimal time slot design for various medium access control (MAC)

procedures for a multiuser scenario is examined for BB LPTV channels.

In this chapter, we demonstrate that much higher data rates, and consequently appreciable

power savings can be achieved when the bit loading algorithm is fully aware of the LPTV

behavior of the channel. The AC mains cycle is divided into microslots, on which bit loading

is handled. It is aware of and exploits the fact that some time microslots have better channel

conditions than others; in practical terms, this suggests that the algorithm views the entire AC

mains cycle when allocating bits and power, instead of allocating for individual time microslots

independently of each other. We address the issue of optimal bit loading by carefully formulating

the integer bit loading problem for LPTV channels and show that greedy-type algorithms yield

20

the maximum data rate. We also develop reduced complexity bit loading schemes that also

handle possible practical considerations such as avoiding peak power levels. Finally, we include

energy efficiency, microslot power level variations, and peak-to-average power ratio (PAPR)

analysis.

In order to motivate our bit loading schemes listed in Section 1.5, we begin in Section 3.4

by computing the improvement in channel capacity from non-adaptive (case #1) to adaptive

schemes (cases #2–4) for various LPTV channels. In Section 3.4.1, we explain and compute bit

allocation for non-adaptation and simplistic adaptation schemes (cases #1–2) using a greedy-

type algorithm. In Section 3.5, we elaborate the optimal LPTV-aware bit loading scheme of case

#3 and we illustrate its improved performance over the simplistic adaptation scheme of case #2;

we also explore the energy efficiency of the proposed bit loading algorithm and the variation

of the microslot power levels over one AC mains cycle. The optimal algorithm is a critical

reference, but it has a certain amount of complexity and it is oblivious to possible constraints

on peak power levels. Because of these issues, in Section 3.6 we develop the sub-optimal reduced

complexity LPTV-aware bit loading with power clipping of case #4 and demonstrate that its

performance is very close to that of case #3. Next, we demonstrate that LPTV-aware bit

loading, complexity reduction and power clipping are also suitable for non-optimal bit loading

algorithms and significant gains in bit rates can be achieved. Two non-optimal bit loading

algorithms that are based on the idea of even distribution of power for a limited number of

good (i.e. low attenuation) subchannels are considered in Section 3.7. Finally, Section 4.6

summarizes our findings.

3.3 Channel Parameters

In this chapter, we employ three commuted channels and three harmonic channels for which

the type, frequency range, [γ]avg, and [Havg] values are shown in Table 3.2. We consider 1 MHz

transmission bands for which, utilizing the model from [11], the periodic time variance is sig-

nificant and matching to those observed in measurements in [14, 15, 8], where channels vary

21

significantly over 2–30 MHz band. The generated channels are comparable to each other in

terms of [γ]avg, and [Havg], and results similar to the ones obtained in the following sections

would be observed for channels in the typical BB PLC 2–30 MHz range with comparable [γ]avg,

and [Havg]. We note that the extent of the LPTV behavior is a function of the loads and

topologies of the power line network. In situations where the LPTV behavior is weak, the

improvements in capacity and bit loading that we present below may not be attainable.

Table 3.1. Channel Parameters

Channel # Type [γ]avg [Havg] Frequency Range (MHz)

1 Commuted 3.62 -42.3 16.5 - 17.5

2 Commuted 3.69 -53.3 16.5 - 17.5

3 Commuted 3.68 -70.2 16.5 - 17.5

4 Harmonic 3.67 -50.6 7.95 - 8.95

5 Harmonic 3.67 -60.6 7.95 - 8.95

6 Harmonic 3.67 -77.6 7.95 - 8.95

3.4 Channel Capacity

For the channel capacity computation, we recognize two cases for LPTV channels: non-

adaptation (case #1) and adaptation (cases #2–4). That is, we compute the data rate achiev-

able taking the communication-system constraints into account and refer to this value as capac-

ity, which is also known as constrained channel capacity. For the non-adaptation case, Hlow,i,

i ∈ {0, . . . , N − 1} is utilized as the transfer function for each microslot when computing the

channel capacity, because designing for the poorest channel is required to guarantee reliable

communication in each microslot. For the adaptation case, on the other hand, distinct mi-

croslot transfer functions Hj(f) are put into use for the capacity calculation in each microslot.

More specifically, considering OFDM with N subcarriers, Hj,i, i ∈ {0, . . . , N−1}, are employed.

We assume the noise is additive white Gaussian noise (AWGN) with N01 power spectral density

1Generalizations to include colored and cyclo-stationary noise, as often experienced in PLC, are straightfor-ward in that their presence can be accounted for by the transfer factors Hj,i. More specifically, if the colored

22

(PSD) of -110 dBm/Hz, where the transmitted signal’s PSD is -55 dBm/Hz. The capacity of

the j-th microslot then can be calculated via

Cj = W

N−1∑i=0

log2 (1 + SNRj,i) , (3.1)

where W is the subchannel bandwidth and SNRj,i is the signal-to-noise ratio (SNR) in the i-th

subchannel of the j-th microslot and is defined as

SNRj,i = Pj,i|Hj,i|2

N0W, (3.2)

and Pj,i is the power allocated for the i-th subchannel of the j-th microslot. For simplicity, for

the capacity considerations in this section, the available power is distributed evenly across all

microslots and subchannels in order to probe the effect of the LPTV behavior.2 Hence, in (3.2),

Pj,i is the same for all subchannels and the only variation is with Hj,i. The average system

capacity for one cycle is then

Cavg =1

M

M−1∑j=0

Cj . (3.3)

For the commuted channel case, Cj takes only two distinct values in (3.1) when calculating Cavg

in (3.3). We denote these two capacities as Chigh and Clow, and δ as the fraction of microslots

with Chigh. By use of these terms, the average system capacity of (3.3) for the commuted

channel case can be expressed simply as

Cavg = δChigh + (1− δ)Clow. (3.4)

and cyclo-stationary noise statistics (variances) are estimated along with the channel gain, they can be incorpo-rated into the bit loading schemes by scaling the channel gain |Hj,i| accordingly. Their presence may change theamount of improvement in bit loading, but if they are considered, the proposed schemes will still be effective.Hence, the following discussion also applies to this generalized case.

2The even distribution of power is not optimal when the actual bit loading is done because of the integerconstraint on bits; however, our capacity results here give strong motivation for the bit loading schemes thatfollow.

23

The above formula assumes a fast transition between the transfer functions of microslots with

Chigh and Clow. In reality with slower transitions, a conservative treatment for reliable commu-

nication is to count the microslots in which the high/low transition is taking place as belonging

to Clow.

Because channel capacity is computed as the average of logarithms [as in (3.1) and (3.3)],

we are also motivated to average the excursion parameter over logarithms, i.e. [γ]avg in (2.5).

In fact, because γj,i is the ratio of |Hj,i| and |Hlow,i|, which are employed in the adaptation

and non-adaptation cases, respectively, [γ]avg is a very meaningful parameter related to the

improvement in channel capacity and bit loading due to adaptation [17].

Figure 3.1. Increase in channel capacity due to channel adaptation for commutedand harmonic channels.

24

The increase in capacity of the six LPTV channels in Table 3.2 due to channel adaptation

is illustrated in Fig. 3.1. We define the increase factor as the ratio of the channel capacity for

the adaptation case to the channel capacity for the non-adaptation case. For a similar [γ]avg

value, the capacity increase is mostly higher for channels with poor channel conditions (i.e.

high attenuation). The increase in capacity for a channel is also non-decreasing for decreasing

transmit signal PSD. This suggests that greater improvement in bit loading can be achieved

for poor channels conditions and reduced transmit signal power levels. We also notice a greater

increase for commuted channels compared to harmonic channels for most cases. The reason for

this behavior is the compactness of the harmonic channels compared to commuted channels for

a given [γ]avg. In other words, the divergence between Hhigh,i and Hlow,i for harmonic channels

is smaller than that for commuted channels.

3.4.1 Greedy Algorithm for Optimal Bit Allocation

The well known water filling algorithm for bit loading maximizes the channel capacity for

band-limited AWGN channels. However, the water filling method is not optimal in the sense

that it does not guarantee integer bit loading. It allows the resultant number of bits to take

non-integer values, which is not practical for a real OFDM system. The necessary and sufficient

conditions for optimal bit allocation in multicarrier modulation systems, and the optimality of

greedy algorithms for bit loading was introduced in [18]. In this work, we follow the approach

in reference [19] to prove the optimality of greedy algorithms for bit loading in LPTV channels.

Consider the following maximization:

max{bi,i=0,...,N−1}

Btot =

N−1∑i=0

bi, (3.5)

subject to three constraints:N−1∑i=0

εi ≤ εtot, (3.6a)

{bi, i = 0, . . . , N − 1} are nonnegative integers, (3.6b)

25

0 ≤ εi ≤ εmaxi , i = 0, . . . , N − 1, (3.6c)

where bi is the number of bits transmitted in the i-th subchannel, Btot is the total number of

bits transmitted in all subchannels, εtot is the total energy available for the OFDM symbol, and

εmaxi is the maximum energy allowed for the i-th subchannel. Water filling solves (3.5) subject

to constraint (3.6a) only, and lets bi take non-integer values. Since all the subchannels have

the same bandwidth, constraints (3.6a) and (3.6c) can also be written using the corresponding

power levels for the total power available for the OFDM symbol, and the maximum allowed

power level for each subchannel accordingly. The optimality of the greedy-type algorithms in

solving (3.5) was shown in [19] to be due to the matroid structure of the considered optimization

problem (See Appendix A and reference [19] for detailed proof). In summary, a greedy approach

is optimal for searching the maximal minimum-weight member of a matroid, and the bit loading

problem in question represents a matroid structure as follows. Let S be the set of all possible

(k, i) pairs, where the (k, i) stands for the k-th bit conveyed in i-th subchannel, and k does

not exceed a certain maximum bmaxi in each subchannel by translating (3.6c) on εmax

i into an

equivalent constraint on bmaxi . Similarly, let I be the collection of all subsets of S, for which the

number of elements in each subset is less than or equal to a certain maximum c, the cardinality

of the subset. Each member of I represents a particular bit allocation pattern for the OFDM

system with total bit transmission not more than c. When the weight for each element (k, i) in

S is defined as the excess energy required to transmit k-th bit in the i-th subchannel, the greedy

algorithm ends up with the bit loading pattern with minimum energy for a given cardinality c.

Since Btot is not known initially for given constraints (3.6a) and (3.6c), the greedy algorithm

starts with Btot = 0, and increments Btot at each step as long as the constraint (3.6a) or (3.6c)

is not violated.

In the context of the LPTV channels, analogous to the above defined problem, we define

the optimization problem for the j-th microslot transfer function Hj(f) as the maximization:

max{bj,i,i=0,...,N−1}

Bj,tot =

N−1∑i=0

bj,i, (3.7)

26

where bj,i is the number of bits allocated to i-th subchannel of the j-th microslot, subject to

the following constraints:N−1∑i=0

Pj,i ≤ P, (3.8a)

{bj,i, i = 0, . . . , N − 1} are nonnegative integers, (3.8b)

where Pj,i is the power allocated for the i-th subchannel of the j-th microslot, and P is the

total power available for each microslot. Here, we have a total power constraint for a microslot,

and the integer bit loading constraint for the maximization problem. Similar to (3.5), for the

(k, i) pair, we formulate the weight function as the excess power required to convey the k-th

bit in the i-th subchannel as

wj(k, i) = Pj,i(k)− Pj,i(k − 1), 1 ≤ k ≤ bmaxj,i , Pj,i(0) = 0, (3.9)

where Pj,i(k) is the power required to convey k bits in the i-th subchannel of the j-th microslot,

and bmaxj,i is the maximum number of bits conveyable by i-th subchannel of the j-th microslot.

For the sake of simplicity, by assuming bmaxj,i to be sufficiently large, we allow bj,i to be as

large as it can be for a subchannel, while maintaining the matroid structure of the problem

and the greedy-type nature of the solution to the maximization in (3.7). In practice, BB PLC

standards allow transmission of up to 10–12 bits per sub-channel [5, 3, 4]. For the generated

LPTV channels, bmaxj,i turns out to be less than 12 bits for most cases including high transmit

power levels (i.e. -55 dBm/Hz). Since bmaxj,i gets smaller for lower data rate tone maps with

reduced power levels, it is less than 12 bits per subchannel at reduced power levels. This leads

to a modified algorithm that can be used for reduced power levels, and is slightly simpler (see

Algorithm 1), which does not put a limit on bmaxj,i . For both (3.5) and (3.7), the greedy-type

algorithm works in an iterative fashion, and at each step, it looks for the (k, i) pair that has

the minimal weight wj(k, i). In other words, it looks for the subchannel that requires the least

amount of power to add one more bit.

This optimization can also be identified as a knapsack problem, where, for a given set of

27

Algorithm 1 Greedy algorithm

1: Set bj,i = Pj,i = 0, i = 0, . . . , N − 1;2: Set Pj,sum = Bj,tot = 0;3: Set ∆Pj,i = wj(1, i), i = 0, . . . , N − 1;4: while (Pj,sum < P ) do5: Find index l such that l = argmin0≤i≤N−1 ∆Pj,i;6: Update Pj,sum = Pj,sum + ∆Pj,l;7: if (P − Pj,sum) ≥ 0 then,8: Set bj,l = bj,l + 1;9: Set Bj,tot = Bj,tot + 1;

10: Update Pj,l = Pj,l + ∆Pj,l;11: Update ∆Pj,l = wj(bj,l + 1, l);12: end if ;13: end while14: exit.

items with a weight and a value, the goal is to fill the knapsack in such a way that the total value

of the knapsack is maximized while the total weight of the knapsack does not exceed a certain

maximum. In this case, the weight of each item is the power required to transmit that item,

the value is the number of bits allocated, and the maximum allowed weight of the knapsack

is the total power available. If the items are designed as (k, i) pairs with a weight of wj(k, i),

then each item has a value of one, and maximizing the value of the knapsack is equivalent to

maximizing the number of items in the knapsack. Intuitively, this can be done by prioritizing

items with less weight, which brings about more items to be placed in the knapsack. This

solution is analogous to the greedy approach, an example of which is the fractional knapsack

algorithm in [8].

The solution to (3.5) is provided in Table 2 of [19], where the algorithm terminates when

the maximum number of bits that can be transmitted due to (3.6c), are conveyed in each

subchannel. This is unnecessary for the solution of (3.7) since (3.7) allows bmaxj,i be as large

as it can be. For the optimization problem in (3.7), we adapted the greedy-like algorithm in

Table 2 of [19] in our implementation of the greedy-like algorithm, for which the pseudocode is

presented in Algorithm 1. In this algorithm, ∆Pj,i represents the power required to transmit

one more bit in the i-th subchannel of the j-th microslot. At each iteration, the algorithm

28

locates the subchannel that requires the least amount of power to add one more bit, adds one

bit to that subchannel, updates power allocated to that subchannel, total power allocated to all

subchannels, and updates ∆Pj,i for that subchannel. For a given minimum symbol error rate

constraint, Pj,i(k) is calculated using the relation for quadrature amplitude modulation (QAM)

symbols [12] as

Pe ≈ 4Q

(√3Pj,i(k)|Hj,i|2N0(k − 1)

), (3.10)

where Pe is the symbol error probability. The relation in (3.15) assumes the ideal case of no SNR

margin, which accounts for the reduction in SNR due to non-ideal issues in the implementation

such as imperfect synchronization. However, this does not affect the comparison of the schemes

in the sections that follow.

3.4.2 Simplistic Adaptation Scheme

We now compare bit loading for LPTV channels for non-adaptation (case #1) and simplistic

adaptation (case #2) schemes using the optimal greedy-type power allocation algorithm of

Section 3.4.1. Parallel to the capacity comparison in Section 3.4, when computing Btot for

a microslot, Hlow,i and Hj(f) are employed as the transfer functions for the non-adaptation

and simplistic adaptation cases, respectively. Also, the increase factor is the ratio of total

number of bits conveyed Btot for the simplistic adaptation case, to Btot for the non-adaptation

case. The increase in raw bit rate due to simplistic adaptation for commuted and harmonic

channels is demonstrated in Fig. 3.2. This agrees with the increase in capacity in Fig. 3.1. More

improvement in bit loading is observed in commuted channels compared to harmonic channels,

and also in channels with higher attenuation compared to channels with lower attenuation. The

improvement also increases with lower transmit signal PSD levels. For Channels 2, 3, 5, and 6,

no transmission is taking place for the non-adaptation case below some transmit signal PSD.

For instance, for Channel 6, the increase factor is infinite for a wide range of low transmit signal

PSDs, where bits are only transmitted for the simplistic adaptation case. In addition, the actual

raw bit rate for the non-adaptation and simplistic adaptation cases in commuted channels are

29

Figure 3.2. Increase in raw bit rate due to the simplistic adaptation schemecompared to the non-adaptation scheme for commuted and harmonic channels.

depicted in Fig. 3.3. The figure indicates that there is a 4–8 dB gain and significant increase

in raw bit rate due to simplistic adaptation in commuted channels. The raw bit rate for non-

adaptation and simplistic adaptation cases in harmonic channels are comparable to the case of

commuted channels, where there is a 4–5 dB gain, and notable increase in raw bit rate due to

simplistic adaptation.

3.5 Optimal LPTV-aware Bit Loading Scheme

In this section, we now formulate the proposed optimal LPTV-aware bit loading scheme for

BB LPTV channels (case #3), and evaluate its performance in comparison with that of the

30

Figure 3.3. Raw bit rate for the non-adaptation and simplistic adaptationschemes for commuted channels.

simplistic adaptation scheme (case #2). We then explore the resultant microslot power levels

and the energy efficiency of the proposed scheme.

3.5.1 Problem Definition

The success of water filling and greedy approaches for bit loading lies in their ability to

favor the subchannels with less channel attenuation in power allocation for an OFDM system,

rather than distributing the power evenly. In the simplistic adaptation case of Section 3.4.2,

this property is exploited via the greedy approach only within each microslot transfer function

Hj(f). Hence, the simplistic adaptation case does not recognize the relative variation of the

transfer function Hj(f) over the AC mains cycle and regards every microslot equal in total

31

power allocation. That is to say, same total power P is allocated for each microslot no matter

how good or bad the channel attenuation is in that microslot. To attain further improvement

in bit loading, we suggest allowing different power levels for each microslot, while maintaining

the average power over the AC mains cycle to be the same as for the simplistic adaptation case.

To accomplish LPTV-awareness and optimal bit loading for LPTV channels, we aggregate

the two-dimensional transfer function, Hj,i, into a large one-dimensional transfer function, which

we call the combined transfer function. The combined transfer function is of size MN , and

includes all the subchannels of all microslots within the AC mains cycle. For the combined

transfer function, we formulate the bit loading problem as maximizing the total bit transmission

over one AC mains cycle as:

max{bj,i,i=0,...,N−1,j=0,...,M−1}

Btot =M−1∑j=0

N−1∑i=0

bj,i, (3.11)

subject to the following constraints:

M−1∑j=0

N−1∑i=0

Pj,i ≤MP, (3.12a)

{bj,i, i = 0, . . . , N − 1, j = 0, . . . ,M − 1} are nonnegative integers, (3.12b)

where constraint (3.12a) is the total power constraint3 over the AC mains cycle, and con-

straint (3.12b) guarantees the integer bit loading. The constraint (3.12a) can also be regarded

as the average power constraint when M is on the left hand side of the inequality. By utilizing

the combined transfer function along with the total power constraint over the AC mains cy-

cle, the system acquires the ability to favor subchannels with less attenuation not just within

a microslot, but across microslots. Moreover, (3.11) has a structure that is similar to (3.7),

with a longer transfer function of size MN , and a total power constraint over the AC mains

3In fact, it is the energy budgets that are added over time, not the power budgets. If the energy budget duringone microslot is E, then it is ME for M microslots. However, since the microslots are of equal duration, and theterm “power” is typically used in the context of loading, we refer to it as a power budget for the AC mains cycleinstead of an energy budget.

32

cycle as MP from (3.12a). Hence, the proof of the optimality of the greedy-type algorithm

of Section 3.4.1 in solving (3.7), is also valid for solving (3.11), with the combined transfer

function and total power MP as its inputs. In other words, the matroid structure of (3.11) can

be proven by the same arguments of Section 3.4.1 and Appendix A, due to the similarity of

the two maximizations (3.7) and (3.11). Thus, the optimality and LPTV-awareness are both

realized for bit loading within one AC mains cycle.

3.5.2 Performance Evaluation

Figure 3.4. Increase in raw bit rate due to the optimal LPTV-aware bit loadingcompared to the simplistic adaptation scheme for commuted and harmonic channels.

In this section, we assess the performance of the optimal LPTV-aware bit loading scheme

compared to the simplistic adaptation scheme. The increase in raw bit rate due to LPTV-aware

bit loading for commuted and harmonic channels is depicted in Fig. 3.4. The increase factor

33

is now the ratio of total number of bits conveyed for the optimal LPTV-aware bit loading case

to the simplistic adaptation case. For all transmit signal PSD values and for all channels, the

increase factor in raw bit rate is greater than one. In other words, the optimal LPTV-aware

bit loading is always slightly or significantly better than the simplistic adaptation scheme,

depending on the transmit signal PSD. This is an anticipated result due to the optimality

of the LPTV-aware bit loading of Section 3.5.1 over one AC mains cycle. In general, there

is more increase in raw bit rate for lower transmit signal PSD values, and for channels with

higher attenuation, which is in line with the results of improvements in channel capacity due

to adaptation, and improvements in bit loading due to simplistic adaptation. Moreover, for

Channel 6, below a certain transmit signal PSD level, no transmission takes place for the

simplistic adaptation case. In this interval, there still is some bit transmission for the optimal

LPTV-aware bit loading case. For this reason, the increase factor is infinite and there is no

data point for this PSD range in Fig. 3.4. Additionally, the actual raw bit rate for the simplistic

adaptation and optimal LPTV-aware bit loading schemes in commuted channels are depicted in

Fig. 3.5. The figure presents that there is up to 6 dB gain due to LPTV-awareness in commuted

channels depending on the channel attenuation and the transmit signal PSD. The raw bit rate

for simplistic adaptation and optimal LPTV-aware bit loading schemes in harmonic channels

are analogous to the case of the commuted channels, where there is up to 2 dB gain due to

LPTV-awareness. The simplistic adaptation is quasi-optimal in these cases for the sample

channels considered.

3.5.3 Resultant Power Levels

The key idea of the optimal LPTV-aware bit loading is to implement an average microslot

power constraint over one AC mains cycle rather than a fixed power level for each microslot,

and to favor microslots with better channel conditions in power allocation in order to improve

bit loading. This causes fluctuation of microslot power levels over one AC mains cycle. This

behavior is illustrated in Fig. 3.6 for Channel 5, where up to 2 dB more power than the average

34

Figure 3.5. Raw bit rate for simplistic adaptation and optimal LPTV-aware bitloading schemes for commuted channels.

is allocated to some of the microslots. In this case, the resultant microslot power levels are at

most -73.0 dBm/Hz, which is 2 dB higher than the average -75 dBm/Hz. As for the resultant

subchannel power levels, the maximum power level is -71.1 dBm/Hz, which is 3.9 dB more

than the average, due to both the greedy algorithm for bit allocation among subchannels and

LPTV-awareness. Furthermore, with an increase factor of 1.05, 5% improvement in bit loading

is achieved compared to the simplistic adaptation case via a small variation in the microslot

power levels, essentially with the same total power consumption over the AC mains cycle. The

microslot power level variation becomes less for higher transmit signal PSD values, where the

distribution is more flat and little gains are achieved in the data rate. For instance, the variation

is less than 0.5 dB for the same channel at -55 dBm/Hz transmit signal PSD, where the raw

35

Figure 3.6. Resultant microslot transmit signal PSD levels for Channel 5, with-75 dBm/Hz average transmit signal PSD over one AC mains cycle, for the optimalLPTV-aware bit loading.

bit rate increase is little, too. On the other hand, the microslot power level variation and

the improvement in the data rate is more for smaller transmit signal PSD values, where some

microslots are not even assigned any power in some cases as a result of the optimal LPTV-aware

bit loading.

3.5.4 Energy Efficiency

In the previous sections, we have demonstrated that significant improvement in bit load-

ing can be accomplished due to simplistic adaptation and optimal LPTV-aware bit loading.

Improvement in bit loading also implies energy saving, since energy per bit transmission is

36

smaller. Additionally, energy saving is a principal design criterion for lower data-rate BB PLC

applications, for which the energy efficiency over a long period is more important than pro-

viding peak data rates. For such cases, transmitting at reduced power levels can be preferable

for two reasons. First, when the transmission power is increased, the channel capacity is also

increased, but only logarithmically. Hence, transmitting at lower power levels and data rates

can be more energy efficient. Second, we can realize more improvement in bit loading due to

simplistic adaptation and optimal LPTV-aware bit loading at reduced power levels as brought

out in the previous sections. To illustrate this, energy per one bit transmission in commuted

and harmonic channels for the case of optimal LPTV-aware bit loading is depicted in Fig. 3.7.

The figure suggests that it is more energy efficient to transmit at lower transmit signal levels,

along with more improvement in bit loading due to LPTV-awareness. However, this comes with

a cost of reduced data rate at reduced power levels as was displayed in Fig. 3.5, which can be

tolerated for lower data-rate BB PLC applications since they usually do not require high data

rates. Reducing power levels also helps reduce interference to other devices, the size of the tone

maps and the encoding complexity at the transmitter; it also reduces the heat dissipation that

could increase the design complexity, and the range for eavesdropping [20].

3.5.5 Power Saving in BB PLC Standards

BB PLC standards IEEE 1901, ITU-T G.hn, and HPGP have mechanisms to reduce power

consumption for devices in the in-home (IH) domain. For instance, IEEE 1901 standard em-

ploys reduced power states, and power management capabilities of stations in order to reduce

the power consumption and electromagnetic emissions [5]. ITU-T G.hn enables low complexity

profile nodes with reduced power consumption levels and reduced data rates. It allows dynamic

and flexible PSD control, where PSD ceiling can be minimized per connection. Furthermore,

devices may have scheduled inactivity inside the AC mains cycle, to reduce the power consump-

tion [3]. Likewise, HPGP specification allows low power modes via reduced duty cycle, in other

words awake times in the power save mode. It utilizes automatic reduction of the transmit

37

Figure 3.7. Energy per one bit transmission for the optimal LPTV-aware bitloading scheme in commuted and harmonic channels.

power as long as reliable communication is achieved, and no hidden nodes are created in the

network. It employs a channel estimation procedure that allows the transmitter to obtain tone

maps for bit loading in various intervals of the AC mains cycle, which can then be used for

channel adaptation [4].

These mechanisms can be combined with the optimal LPTV-aware bit loading to reduce

power consumption of BB PLC devices. The low data rate requirement for lower data-rate BB

PLC allows devices to operate at reduced power levels, where the improvement in bit loading

due to LPTV-awareness is high, and energy per bit transmission is low. Additionally, power save

modes allow some microslots to be idle for transmission. As discussed earlier in Section 3.5.3,

for reduced power levels, some microslots may not be allocated any power for transmission as

38

a result of the bit loading algorithm. This behavior is illustrated in Fig. 3.6, which can be used

when scheduling sleep and awake times for the lower data-rate BB PLC devices for minimizing

power consumption and maximizing the raw bit rate in the power save modes.

3.6 Sub-optimal Reduced Complexity LPTV-aware Bit Loading

In this section, we apply two modifications to the optimal LPTV-aware bit loading with the

purpose of reducing complexity and microslot peak power, respectively. The optimal LPTV-

aware bit loading maximizes the raw bit rate over the AC mains cycle, but it also increases the

system complexity since the algorithm has to work on a long combined transfer function of size

MN . Because of this drawback, the sub-optimal reduced complexity LPTV-aware bit loading

with power clipping scheme (case #4) initially limits the problem to only allocating M microslot

power levels, Pj . This requires the algorithm to work on a much smaller data set of size M

initially, compared to MN , and only store M microslot power levels in the end, compared to

MN subchannel bit and power values. Once the algorithm finds M microslot power values,

the computational complexity of the system is the same compared to the simplistic adaptation

case during the AC mains cycle.

3.6.1 Reduced Complexity LPTV-aware Bit Loading

In order to determine Pj , the transfer function for the j-th microslot, Hj,i, 0 ≤ i ≤ N − 1,

is represented by (i.e. reduced to) a single value denoted as Hj . By doing so, the algorithm

computes {Pj} to maximize the transmission rate. Then {Pj} is stored for use during the AC

mains cycle. The other important step is to properly normalize the input power level seen by

the algorithm, so that (3.15) works properly, and thus P = PM/N is used in the algorithm.

The proposed reduced complexity LPTV-aware bit loading works as follows:

1. Represent each microslot by a single value Hj .

2. Compute Pj , an N -normalized microslot power allocation, using the algorithm in Sec-

tion 3.4.1 with Hj and P as inputs.

39

3. Compute the final distribution Pj as

Pj =Pj∑M−1j=0 Pj

PM. (3.13)

where the normalization by N is undone and possible rounding errors are eliminated.

4. During the j-th microslot of the AC mains cycle, use the algorithm in Section 3.4.1 to

allocate Pj for the subchannels in that microslot, Hj,i.

Because the algorithm works on Hj first, instead of Hj,i in its entirety, it reduces the complexity

by a factor of N .

We propose two ways of determining Hj . The first method is to use the mean magnitude

value, the maximum magnitude value, or a weighted sum of the two, which works well for mildly

frequency-selective channels such as the ones in Table 3.2. However, for channels in the 2–30

MHz frequency band, a more elaborate approach is needed due to the highly frequency selective

channels. For these channels, we propose to select a representative channel gain for a microslot

such that its associated capacity is equal to the average subchannel capacity of that microslot,

for a given transmit signal PSD. That is, we choose |Hj | according to:

CjN

= W log2

(1 + Pj,i

|Hj |2

N0W

). (3.14)

3.6.2 Power Clipping

Since the optimal LPTV-aware bit loading permits the variation of microslot power levels,

when operated at close vicinity of maximum power levels, the resultant microslot power levels

may exceed a certain maximum allowed power level, which is not a concern when operated at the

reduced power levels for energy efficiency. In the following we describe a simple power clipping

procedure for reduced complexity LPTV-aware bit loading such that maximum microslot power

levels are not exceeded. Given the final power distribution for microslots Pj , and denoting the

maximum power level by Pmax, proceed as follows:

40

1. Limit the power assigned to microslots exceeding Pmax to Pmax.

2. Look for the highest power value that is less than Pmax among the remaining power levels.

3. Add the excess power remaining due to clipping to this microslot, and check if the power

for this microslot is less then Pmax:

(a) If Yes, terminate the algorithm.

(b) If No, clip the power for this microslot at Pmax, and go back to Step 2.

Here, because the bit loading algorithm favors the microslots with relatively better conditions,

they are once more given preference at Step 2.

Figure 3.8. Raw bit rate for the sub-optimal reduced complexity LPTV-awarebit loading with power clipping, and the optimal LPTV-aware bit loading schemesin commuted and harmonic channels.

41

3.6.3 Performance Results

In this section, we compare the performance of the sub-optimal reduced complexity LPTV-

aware bit loading with power clipping (case #4) with the optimal LPTV-aware bit loading

of Section 3.5.1 (case #3). The raw bit rate results for commuted and harmonic channels

are depicted in Fig. 3.8. For all channels and transmit signal PSD levels, the sub-optimal

reduced complexity LPTV-aware bit loading with power clipping performs very close to the

optimal LPTV-aware bit loading, which makes it an attractive solution for the problem of

bit loading in LPTV channels. For these results, we have picked the maximum magnitude

values for determining Hj , which is a good choice for especially low input transmit signal PSD

levels for mildly frequency-selective channels. For the low transmit signal power levels, only

the subchannels with high |Hj,i| qualify to transmit, and in this case the maximum magnitude

values for Hj are good representatives of these channels. For highly-frequency selective channels

in 2–30 MHz range, when Hj are chosen according to (3.14), the reduced complexity Scheme 4

achieves raw bit rates close to the optimal Scheme 3, and outperforms Scheme 2 for the most

transmit signal PSD levels, similar to the results illustrated in this chapter.

As for the power clipping, the resultant microslot power levels of the reduced complexity

LPTV-aware bit loading are assured not to exceed a certain power level Pmax. For the results in

Fig. 3.8, we choose Pmax such that the PSD of -55 dBm/Hz is not exceeded (e.g. the transmit

spectrum mask, Figs. 13–20, IEEE 1901 [5]). In general, when the average and the maximum

transmit signal PSDs are equal, no improvement can be achieved exploiting the LPTV behavior

if the power clipping is in place. This is the case where the transmit signal PSD is in the close

vicinity of -55 dBm/Hz. However, when the average transmit signal PSD is several dB lower

than the maximum allowed transmit signal PSD, as is the case for the reduced transmit signal

power levels, the power clipping is not needed and the resultant microslot power allocation Pj

is not adjusted by power clipping.

Additionally, the gains in raw bit rates due to the proposed schemes in previous sections

would also be observed when error-correcting codes are utilized, since coding can similarly be

42

applied regardless of the specifics of the bit loading scheme employed.

3.6.4 Peak-to-average Power Ratio

We also examined the peak-to-average power ratio (PAPR) of the transmitted time domain

OFDM signal, which is a concern for OFDM systems. OFDM systems may suffer from high

PAPR, which affects the performance of the power amplifier at the transmitter. We compared

the PAPR for Schemes 2 and 4. The results indicate that the PAPR is about the same for

the two schemes, which also results in a similar quantizing noise. We considered PAPR for

each microslot, as well as the PAPR over the whole AC cycle. For both cases, the difference

between the PAPRs for Schemes 2 and 4 is less than 0.5 dB for all ranges of transmit signal

PSDs. For some cases, the PAPR for Scheme 4 is smaller than Scheme 2, and for some cases,

it is larger. For very low transmit signal PSDs, the PAPR for Scheme 4 can be up to 9 dB

more than Scheme 2. This is due to the idle microslots, which results in a much lower average

power. However, when the average power is computed considering only non-idle microslots, the

PAPR is again about the same as Scheme 2. In general, the PAPR over the whole AC cycle

is between 9–11 dB, which is about 2 dB larger than the PAPR computed by averaging the

PAPR of individual microslots for both schemes.

3.7 LPTV-aware Bit Loading for Non-optimal Algorithms

The ideas presented in this chapter such as LPTV-aware bit loading, complexity reduction

via representative values, and power clipping can be applied to not only the optimal greedy-type

bit loading algorithms, but also to other non-optimal bit loading algorithms. In this section, we

implement LPTV-aware bit loading and its reduced-complexity version with power clipping on

two other algorithms that utilize even-like power distribution [17, 21]. The algorithms presented

here are not optimal in terms of bit loading, but similar improvements in bit rate can be achieved

by utilizing LPTV-aware bit loading approach.

43

3.7.1 Algorithm A

Algorithm A simply allocates power evenly for subchannels that can support a signal con-

stellation of size Ki = 4 according to a given error-rate criterion (cf. [12]). Hence, for a given

total power and a minimum symbol error rate constraint, Algorithm A is as follows:

1. Allocate power evenly to all subchannels.

2. For subchannels that cannot support Ki = 4, and a target error rate according to (3.15),

set power to 0.

3. Allocate the total power evenly to remaining subchannels.

4. Compute unquantized Ki for the best channel using (3.15)

Pe ≈ 4Q

(√3Pi|Hj,i|2N0(Ki − 1)

). (3.15)

5. Quantize Ki for that subchannel.

6. Compute power required for that subchannel and update the remaining unallocated power.

7. Return to Step 3 and continue until all subchannels are assigned a bit and power value.

In this algorithm, Ki is allowed to be as large as it can be to support the desired Pe.

3.7.2 Algorithm B

In this section, we describe Algorithm B, which we obtained by modifying Algorithm A with

some enhancements. Steps 1 and 2 of Algorithm A disqualify many subchannels of the transfer

function based on the average transmit signal PSD. Since the algorithm does not take into

account the possibility that excess power might be available for subchannels with moderate

channel conditions, it forces the algorithm to allocate as many bits as possible to a limited

portion of the spectrum. Algorithm B avoids this shortcoming in that disqualifies only one

subchannel at a time. After a subchannel is disqualified, it updates the average power level

44

available for the remaining subchannels and discards the next subchannel based on this new

increased power level. Algorithm B works as follows:

1. Allocate power evenly to all subchannels. Assume all subchannels are in the list of quali-

fying subchannels.

2. Find the worst subchannel among qualifying subchannels, check if it supports transmission

with Ki = 4:

(a) If Yes, go to Step 3.

(b) If No:

i. Delete that subchannel from qualifying subchannels list.

ii. Allocate power evenly to remaining qualifying subchannels. Go back to Step 2.

3. Allocate the total power evenly to qualifying subchannels.

4. Follow Steps 4–7 of original LPTV-aware bit loading.

3.7.3 Performance Analysis


Channel # Type [Havg] [γavg]

1 Commuted -42.3 3.62

2 Commuted -53.3 3.69

3 Commuted -70.2 3.68

4 Harmonic -50.6 3.67

5 Harmonic -60.6 3.67

6 Harmonic -77.6 3.67

In this section, we analyze the performace of the non-optimal Algorithms A and B. For

both algorithms, we choose 10−5 as the bit-error-rate (BER) requirement, -55 dBm/Hz as the

average power spectral density (PSD) of the transmit signal, and -110 dBm as the noise PSD

45

as practical values for Equation (3.15). Also for all cases, there are N = 256 subchannels and

M = 50 microslots in the OFDM system. The parameters for the channel generated for analysis

are listed in Table 3.2.

Figure 3.9. Algorithm A LPTV-aware, Algorithm B LPTV-aware, and Algo-rithm B reduced complexity LPTV-aware bit loading with power clipping using themaximum magnitude values for Hj for commuted channels.

The results for the comparison of: (i) Algorithm A LPTV-aware, (ii) Algorithm B LPTV-

aware, and (iii) Algorithm B reduced complexity LPTV-aware bit loading with power clipping,

for commuted and harmonic channels are depicted in Figs. 3.9 and 3.10, respectively. For

both commuted and harmonic channels, Algorithm B LPTV-aware bit loading outperforms

the Algorithm A LPTV-aware bit loading in all cases. Also, Algorithm B reduced complexity

46

LPTV-aware bit loading with power clipping performs very close to Algorithm B LPTV-aware

bit loading in all cases, and outperforms the Algorithm A LPTV-aware bit loading in most

cases. For only a few input transmit signal PSD levels, Algorithm B reduced complexity LPTV-

aware bit loading with power clipping performs 2–3% worse than Algorithm B LPTV-aware bit

loading. Our results show that using the maximum magnitude values for Hj is a good choice

compared to the mean value. When the input transmit signal PSD is low, only the subchannels

with high |Hj,i| qualify to transmit, and in this case the maximum magnitude values in Hj

are good representatives of these channels. Also, when a weighted sum of the mean and the

maximum value is chosen for Hj , the improvement is in between the two improvements when

the maximum is used alone and the mean is used alone.

For all channels, we also observe peaks in the increase factor curves, beyond which no

transmission is taking place because the SNR is too low. These are the power levels that

only a limited spectrum of a portion of the microslots can support transmission with channel

adaptation or Algorithm A LPTV-aware bit loading algorithm. For instance, for Channel 4

with -86 dBm/Hz input transmit signal PSD, for the channel adaptation case, Algorithm A

disqualifies 42 of the microslots for transmission based on the given power level. With Pj = P ,

5–6 bits in only 14–24 subchannels are transmitted in the eight remaining microslots. With

Algorithm A LPTV-aware bit loading, the same spectrum is initially disqualified, but the

excess power from 42 microslots is transferred to these eight microslots and eight bits are now

transmitted in these same subchannels, which results in 1.50 increase factor. However, with

Algorithm B LPTV-aware bit loading algorithm, since each time it disqualifies subchannels

based on an updated power level, 24 microslots can support transmission. In each of these

24 microslots, 2–3 bits are transmitted in 157–256 subchannels. This better utilization of the

available spectrum results in the peak improvements in all of the six LPTV channels.

When power clipping is in place, the algorithm forces the Algorithm B reduced complexity

LPTV-aware bit loading not to exceed a certain power level Pmax. For the results, we choose

Pmax such that the PSD of -55 dBm/Hz is not exceeded. For instance at -55 dBm/Hz transmit

47

Figure 3.10. Algorithm A LPTV-aware, Algorithm B LPTV-aware, and Algo-rithm B reduced complexity LPTV-aware bit loading with power clipping using themaximum magnitude values for Hj for harmonic channels.

PSD for Channel 3 and 6, Pj is forced to be the same for all microslots, and an increase factor

of 1.32 and 1.42 is only due to the deployment of the enhanced algorithm as shown in Figs. 3.11

and 3.12, respectively. Still at this power level, Algorithm B reduced complexity LPTV-aware

bit loading with power clipping performs better than Algorithm A LPTV-aware bit loading

due to the enhanced algorithm. When the average and the maximum transmit signal PSDs

are equal, no improvement can be achieved exploiting the LPTV behavior. However, when the

average transmit signal PSD is lower than the maximum transmit signal PSD, as is the case

for the rest of the improvement range for Channel 3 and 6, and for the remaining channels, the

48

power clipping does not change Pj because Pj does not exceed Pmax.

Figure 3.11. Simplistic channel adaptation with Algorithm B, and Algorithm BLPTV-aware bit loading for commuted channels.

Additionally, because Algorithm B LPTV-aware bit loading uses a modified algorithm and

applies it to the combined transfer function, large increase factors we observe are due to both

the enhanced algorithm, and the LPTV awareness of the algorithm. To see the effect of LPTV-

awareness only, we also examined the simplistic channel adaptation case using Algorithm B. In

this case, Pj = P , and Algorithm B is employed for all microslots. Hence, the improvement is

due to the utilization of Algorithm B only. As shown in Figs. 3.11 and 3.12, significant increase

factors are achieved due to LPTV-awareness in commuted and harmonic channels, respectively.

However, for harmonic channels, the improvement due to LPTV-awareness is smaller than that

49

of the commuted channels, which is a similar result to the case for the Algorithm A LPTV-aware

bit loading.

Figure 3.12. Simplistic channel adaptation with Algorithm B, and Algorithm BLPTV-aware bit loading for harmonic channels.

3.8 Summary and Conclusions

In this chapter, we examined the problem of optimal bit loading in LPTV channels for

BB PLC. We demonstrated that a significant increase in channel capacity can be attained by

adaptation to the LPTV channel, and the greedy-type algorithms are optimal for bit loading,

when the combined transfer function and an average power constraint over one AC mains cycle

is utilized. In addition, it can be more energy efficient to transmit at reduced power levels,

50

which is also the regime where the most significant gains in raw bit rate are achieved due to the

proposed LPTV-aware bit loading. Next, we presented that when representative values from

the microslot transfer functions are employed, the computational complexity of the system is

greatly reduced, while at the same time preserving nearly all of the gains in bit loading. Finally,

power clipping algorithms can be practical in avoiding peak power levels for the case of LPTV-

aware bit loading schemes, since they allow for variation of microslot power levels. The ideas

presented in this chapter are also applicable to non-optimal bit loading algorithms, for which

similar increase in bit rates are observed. All of our contributions have the aim of saving power

and improving bit loading appreciably for BB PLC systems with little additional complexity.

The proposed schemes are suitable for both multimedia and SG-type applications, and also

applicable to BB PLC standards such as IEEE 1901, ITU-T G.hn, and HPGP.

51


52

Chapter 4

Channel Estimation

4.1 Key Points of the Chapter

• Incline-type pilot arrangement can be utilized for BB PLC channels to help reduce the

interpolation error inherent in comb-type pilot arrangement.

• Block-type pilot arrangement can be utilized for CE, for which the overhead due to pilot

symbols is high. On the other hand, decision directed approaches require less overhead,

but their ability to track sudden changes in the channel is limited.

• The difference between the channel estimates of two consecutive OFDM symbols—when

analyzed in a transform domain (TD)—are different in their frequency content. A change

in the transfer function appears as an increase in the low frequencies, and the presence of

impulsive noise appears as an increase in all frequencies in TD.

• The proposed final scheme utilizes pilot symbols placed widely apart in order to extract

information about the changes in the transfer function and the power line noise via TD

analysis, and exploits this information for switching between various CE schemes.

• The overhead of the proposed scheme for CE is low, and sudden changes in the channel

are tracked effectively. Therefore, the effects of the LPTV channel and the impulsive noise

on CE are mitigated.

53

4.2 Introduction

In Chapter 3, we developed optimal and reduced complexity LPTV-aware bit loading

schemes for BB PLC orthogonal frequency division multiplexing (OFDM) systems to improve

data rates for LPTV channels. In a practical setting, channel estimation (CE) must be done

prior to bit loading for these algorithms to work properly. In this chapter, we study pilot-based

and decision directed CE schemes and develop a robust CE method based on TD analysis

approach.

One way to estimate the channel is to transmit a sequence known by the receiver. This

is called pilot based (data-aided, supervised, or trained) CE, which is one of the widely used

OFDM CE techniques. In this approach, pilot symbols are inserted into the OFDM symbol

at certain subcarrier locations for CE. When all subcarriers carry pilot symbols (i.e. block-

type pilot geometry), no interpolation is needed, but estimation overhead due to pilots is high.

Instead, the density of the pilots can be decreased, with the cost of increase in interpolation

error [22].

In [13], non-adaptive least squares (LS) and TD algorithms are used initially to estimate

the channel at pilot locations for an LPTV channel in the presence of power line impulsive

noise. Different interpolation techniques are used to estimate the channel for the remaining

subchannels, and their performances are evaluated. The tradeoff between channel estimation

accuracy and system throughput is studied in [23] for a Multi-Input Multi-Output (MIMO)

OFDM system for in-home PLC. In [24], a comparison has been made for three pilot based

channel estimation schemes for the indoor power line: LS, LMS, and linear minimum mean

squared error (LMMSE), and in [22], different interpolation techniques (linear, cubic, B-spline,

second order Gaussian, DFT [zero padding], and nearest-neighbor) for the pilot based approach

are compared for the power line channel.

In this work, we first focus on various pilot arrangement schemes with the goal of reducing

interpolation error, and providing better CE with low overhead and better channel tracking

capability. Different from the previous work in the literature, we study incline-type pilot ar-

54

rangements and show that these compare favorably against block-type and comb-type (these

pilot arrangement schemes are illustrated later in Fig. 4.2). LS and LMMSE estimators with

linear and cubic interpolation are implemented for pilot-based CE.

Another method for CE is the decision directed approach, where the channel estimate from

the previous symbol is used to decode the symbols, and these decisions are then utilized to

update the channel estimate. In this approach, pilot symbols can be used to obtain an initial

estimate, but then the changes in the channel are tracked by the decisions with no pilot symbol

assistance. Hence, the estimation overhead is lower compared to the pilot based approach.

However, if the channel changes abruptly, the noise level is high, or the channel is poor (high

signal attenuation), the decisions can become highly inaccurate. This may result in errors in

decoding the symbol, which also results in high estimation error.

Lastly, in addition to the LPTV channel, one other challenge for CE in BB PLC is the

impulsive noise caused by the switching events in the power line network. Impulsive noise

typically has large amplitudes compared to the background noise and it has a short duration [1].

In this chapter, after analyzing various pilot geometry for CE, we develop a transform

domain (TD) analysis approach for BB PLC channels to mitigate the effects of the LPTV

channel and the impulsive noise via pilot symbols placed widely apart. TD analysis examines

the difference between channel estimates of two consecutive OFDM symbols in a transform

domain, and detects changes in the estimate due to a change in the transfer function or noise

characteristics. We then exploit the information captured by TD analysis and develop a robust

CE scheme for BB PLC channels that utilizes various CE schemes based on the outcome of TD

analysis. Our method is different than conventional schemes in the literature, which are based

on a single approach (e.g. pilot-based with certain pilot geometry or decision directed CE) and

lack the ability to take advantage of employing different schemes for different scenarios that

involves abrupt changes in the channel and the noise. The proposed final scheme has small

overhead and high estimation accuracy, and is able to track changes in the channel transfer

function and the noise via TD analysis.

55

The remainder of this chapter is as follows. In the next section, we analyze pilot-based

CE with different pilot geometry for an OFDM system, explain and compare LS and LMMSE

schemes for CE from different perspectives, and in Section 4.4 we develop the TD analysis for

CE. In Section 4.5, we investigate various cases for CE, evaluate the performance of various

schemes, and explain the proposed final CE method based on this analysis. Finally, Section 4.6

summarizes our findings.

4.3 Channel Estimation

4.3.1 LS and LMMSE Estimators

For the OFDM system described in Section 2.2.1, the LS solution for CE is simply:

hLS = X−1y. (4.1)

For the comb-type and incline-type pilot geometries, interpolation is performed for estimating

the channel for non-pilot subcarriers. The LS algorithm is simple to implement, since it requires

only a division operation at the receiver, and it does not require any statistical information

about the channel and the noise. Another estimator is based on LMMSE, which is attained by

application of the Wiener-Hopf equation [25]:

hLMMSE = RhyRyy−1y, (4.2)

where Rhy is the cross correlation matrix:

Rhy = E[hyH ] = E[h(Xh + n)H ]

= E[hhHXH + hnH ]

= RhhXH , (4.3)

56

and (· · · )H is the Hermitian operator, and E[hnH ] is zero when the channel and the noise are

assumed to be uncorrelated. Similarly, the autocorrelation of the received symbol y can be

formulated by:

Ryy = E[yyH ] = E[(Xh + n)(Xh + n)H ]

= E[XhhHXH + XhnH + nhHXH + nnH ]

= XRhhXH + E[nnH ] = XRhhXH + σ2nI, (4.4)

where I is the N ×N identity matrix, and σ2n is the variance of AWGN noise. By substituting

equations (4.3) and (4.4) into (4.2), we obtain

hLMMSE = RhhXH(XRhhXH + σ2nI)−1y. (4.5)

As seen in equation (4.5), in order to compute hLMMSE, the channel autocorrelation function

and the variance of the power line noise must be known. It also requires a matrix inverse

operation that increases the complexity of the scheme. There have been several studies in the

literature that focus on reducing the complexity of the LMMSE algorithm, which is out of the

scope of this work. LMMSE outperforms LS provided that, the Rhh and σ2n are accurate. In

the case where there is a mismatch between the actual values and the estimated values for

these parameters, LMMSE can result in worse performance than LS depending on how big

the error is for these parameters. In practice for the receiver, computing σ2n is relatively easy,

since it would be sufficient to compute the noise variance simply when the channel is idle.

However, computing Rhh may introduce some problems. First, Rhh is usually different for

different frequency bands, so if the frequency band of communication changes, Rhh must be

recomputed. Second, in practice, if it is computed by averaging hhH over time to replace the

expectation operation, a sufficient number of distinct channel transfer functions must be used

for estimating Rhh, such that the error is small.

In order to find the number of distinct transfer functions needed to accurately estimate

57

Rhh, we first computed Rhh by averaging 1000 distinct transfer functions using the model

described in Section 2.1. This being the actual value for Rhh, we computed the estimate Rhh

for different number of distinct transfer functions, and compared it to the actual Rhh. The

channels generated here correspond to random topologies, where the length of the line segments

are from a random uniform distribution from 1–40 m. The results for the average of 50 trials

are illustrated in Fig. 4.1. Our results indicate that using more transfer functions reduces the

mismatch as anticipated, and for less than 50 distinct transfer functions, the error can be large,

and does affect the performance of the LMMSE estimator.

Figure 4.1. Normalized mean square error for Rhh computation using differentnumber of distinct transfer functions.

4.3.2 Pilot-based Channel Estimation

4.3.2.1 Pilot Arrangement

For pilot-based CE, various schemes can be utilized. For the pilot-based schemes, the trans-

mitted symbols at certain subcarriers are known by the receiver. Different pilot geometry can

be utilized for CE as illustrated in Fig. 4.2. For the block-type pilot geometry, all subchannels

58

subc

arrie

r in

dex

pilot

data

.subc

arrie

r in

dex

OFDM Symbolindex

.

(a) (b) (c)

OFDM Symbol index

.

.

subc

arrie

r in

dex

.

.

.

OFDM Symbol index

(d)

subc

arrie

r in

dex

.

.

.

OFDM Symbol index

Figure 4.2. Different types of pilot arrangement: (a) block-type pilots; (b) comb-type pilots; (c) comb-type pilots where pilots are placed widely apart; (d) incline-type pilots

carry pilots. In Fig. 4.2a, one out of three OFDM symbols carry pilots1, but more or less

frequent pilot symbols can be utilized based on how fast the channel changes and how much

estimation overhead can be tolerated. For the comb-type pilot arrangement, pilots are inserted

at some subcarriers with certain density, and data is transmitted for the non-pilot subcarriers.

In Fig. 4.2b, one out of L = 3 subchannels carry pilots and the remaining subchannels carry

data. In Fig. 4.2c, one out of L = 10 subchannels carry pilots, where the pilots are placed

widely apart to reduce the estimation overhead. We utilize this idea of low density pilots later

in this chapter in order to be able to perform TD analysis for CE and in the meantime keep

the estimation overhead low. Using the same number of pilots, the comb-type arrangement

has the advantage of tracking changes in the channel over time. For BB PLC LPTV channels,

there can be many channel realizations within one AC cycle of the mains [11], for which the

pattern would repeat itself for some period of time. In order to estimate the LPTV channel, if

1For the block type arrangement, it is assumed that the channel is constant for the non-pilot OFDM symbolsand there is a tradeoff between estimation accuracy and the amount of overhead required. For LPTV channels,we assume that the channel is constant for one OFDM symbol time and the channel can change from one OFDMsymbol to another. For this reason, in order to track changes in the channel, when block-type arrangement ismentioned in this paper, we mean that all of the OFDM symbols carry only pilots for at least the duration ofone AC mains cycle [16].

59

OFDM symbols carrying only pilots are transmitted for one full AC cycle, the overhead will be

high compared to the comb-type arrangement [16]. Therefore, a comb-type arrangement can

be preferable to reduce the overhead with some increase in complexity.

Figure 4.3. Normalized mean square error using linear interpolation, L = 5.

When the comb-type arrangement is deployed, CE is done at pilot symbol locations first.

Then, CE for the remaining subchannels is achieved by interpolation. For these subchannels,

the difference between the actual channel response and the value obtained by interpolation is the

estimation error, part of which is caused by interpolation (See Appendix B and reference [22]).

This is due to the fact that the estimates in these subchannels rely on the estimates at pilot

locations and interpolation. For the comb-type arrangement with a slow fading channel, the

estimates for the pilot locations can become more accurate by averaging over many OFDM

60

symbols. However, the interpolation error would still remain. The interpolation error can also

be reduced by increasing the density of pilots, which then increases the estimation overhead. In

this section, we investigate the case when the pilot locations are shifted each time a new OFDM

symbol is transmitted (incline-type [26]) for the BB PLC channel, in comparison with block-

type and comb-type arrangement. Incline-type pilot arrangement is illustrated in Fig. 4.2d. To

the best of our knowledge, this is the first study that exercises incline-type pilot arrangement

for the BB PLC channel.

Figure 4.4. Normalized mean square error using linear interpolation, L = 10.

For the incline-type arrangement, unlike the comb-type, the channel estimates for non-pilot

subcarriers do not rely solely on the interpolated values. Since each subcarrier carries both pilot

and data symbols, the estimate is based on both pilot based estimation and interpolation. For

61

comparison, we assume that the channel does not change for L OFDM symbols, and consider

the following schemes:

• Scheme A: Comb-type arrangement is utilized, and L channel estimates at pilot locations

are averaged. Then, one time interpolation is performed to obtain channel estimates for

the non-pilot subcarriers.

• Scheme B: Comb-type arrangement is utilized, and for each OFDM symbol, interpolation

is performed. Then, the average is computed for all subchannels.

• Scheme C: Incline-type arrangement is utilized, and no interpolation is performed. The

one time pilot estimates are put together to form the estimate for the whole OFDM

symbol.

• Scheme D: Incline-type arrangement is utilized, and for each OFDM symbol, interpola-

tion is performed. Then, the average is computed for all subchannels.

• Scheme E: Block-type arrangement is utilized for one out of L OFDM symbols, and no

interpolation is performed.


Channel # [havg]

1 -19.1

2 -22.3

3 -28.3

4 -34.4

5 -41.2

6 -46.3

We also note that the first and the last suchannels are always assigned pilots in all schemes,

and the number of pilots utilized for L OFDM symbols are nearly the same for all schemes.

Therefore, the overhead for each scheme is approximately the same. We examined LS and

62

Figure 4.5. Normalized mean square error using cubic interpolation, L = 5.

LMMSE estimators with linear and cubic interpolation options for L = 5, and L = 10. For per-

formance evaluation, we modified six channels for which the attenuations are given in Table 4.1,

in order to generate channels from low to high attenuation range. 50 trials are computed for

each channel at each attenuation level, and the average of the six is computed. To compute the

normalized mean square error, the actual channels are normalized, and the estimates are scaled

accordingly. The results are shown in Figs. 4.3–4.6, where the LMMSE estimator outperforms

LS, and cubic interpolation outperforms linear interpolation in most cases. Schemes C and E

perform the same and they are not affected by the density of pilots. This is because Scheme

C is equivalent to Scheme E when the channel and noise characteristics are assumed to be the

same for L OFDM symbols. We also observe that Scheme D (incline-type with interpolation)

63

outperforms Schemes A and B for the most part, or performs very close for the remaining part.

It also outperforms Scheme E for almost all cases except for linear interpolation with L = 5

in the low attenuation (high signal-to-noise ratio (SNR)) region. All schemes perform close to

one another and do well in the high SNR region. We notice that the benefit due to incline-type

pilot arrangement with interpolation (Scheme D) decreases as the pilot density decreases for

the linear interpolation case, since linear interpolation suffers when the pilots are too distant

from one another. On the contrary, cubic interpolation is more immune to sparsely located

pilots, and is not affected as much as the linear interpolation case. Overall, the results indicate

that Scheme D is a good candidate for LPTV channels especially in the low SNR scenarios,

where estimation error is large due to high attenuation and interpolation error. We note that

the improvement in CE due to the incline type pilot arrangement depends on the interpolation

kernel being used, and can vary based on the choice of the interpolation kernel.

4.3.3 Decision-directed Approach

In addition to pilot-based CE explained in the this section, another method for CE is the

decision directed approach. This minimizes the overhead for CE by depending on the decisions

made at the receiver without the knowledge of the transmitted symbol. An initial estimate

can be achieved using pilots, but then no pilots are utilized. The channel estimate from the

previous OFDM symbol is used to decode the current OFDM symbol, then with those decisions,

the channel estimate is updated for the current OFDM symbol and used to decode the next

OFDM symbol. This method has smaller overhead compared to pilot-based approaches, but

for the LPTV channel, when the channel changes abruptly, the decisions based on the previous

estimate are highly erroneous. This results in large estimation error and additional decoding

errors. When the channel remains constant, the signal attenuation is low, and the impulsive

noise is not present, the decisions are more dependable and this method can be preferred to

the pilot-based approaches due to the lower overhead. However, for the BB PLC channel, the

channel and the noise may change abruptly and the signal attenuation can be high. Therefore,

64

Figure 4.6. Normalized mean square error using cubic interpolation, L = 10.

both the pilot-based and decision directed CE approaches have some drawbacks for the BB

PLC channel due to the LPTV channel characteristics and the impulsive noise.

4.4 Transform Domain Analysis

Inspired by the work in [27, 13], in this section, we explain TD analysis that we propose for

CE in BB PLC channels in the presence of power line impulsive noise [28]. The goal of the TD2

analysis is to identify the cause of a change in our channel estimates. If the receiver has the

knowledge of whether a change in the estimate is due to a change in the transfer function, the

presence of impulsive noise, or both, the CE algorithm can exploit this information to improve

2The transform domain is the Fourier transform of data that has already been Fourier transformed.

65

Figure 4.7. TD analysis for a single realization of four cases for LS estimates

its estimate. TD analysis proposes that the changes in the estimates will be different in their

frequency content (TD), when they are caused by a change in the channel transfer function, or a

change in the noise level. Let us assume that the receiver has a current estimate for the channel

h1, and is receiving a new OFDM symbol to compute a new estimate h2 for the channel. Four

cases can be classified for the power line:

• Case A: The channel remains the same (h2 = h1). Background AWGN is present in the

received OFDM symbol.

• Case B: The channel remains the same (h2 = h1). Background AWGN and impulsive

noise are present in the received OFDM symbol.

66

Figure 4.8. Low frequency metric for Cases A–D.

• Case C: The channel has changed (h2 6= h1). Background AWGN is present in the

received OFDM symbol.

• Case D: The channel has changed (h2 6= h1). Background AWGN and impulsive noise

are present in the received OFDM symbol.

In order to find out which of the above cases has occurred, TD analysis can be utilized: ex-

amining the frequency content of the change in the channel estimate via the following N point

DFT:

∆hTD = DFT (|h2| − |h1|). (4.6)

We illustrate |∆hTD| of one channel realization of h1 and h2 for Cases A–D in Fig. 4.7. Four

apparent patterns are observed. For instance, for Case B compared to Case A, we notice an

increase in all frequencies. For Case C compared to Case A, an increase in the edges of the

67

Figure 4.9. High frequency metric for Cases A–D.

spectrum is observed, which corresponds to low frequencies. And for Case D compared to Case

A, a further increase in the edges for low frequencies, and an increase similar to Case B for

the middle of the graph for high frequencies is recognized. The general trend is as follows: a

change in the transfer function appears as an increase in the low frequencies, and the presence

of impulsive noise appears as an increase in all frequencies. TD analysis motivates the following

metrics to quantify these changes: the low frequency metric γLF and the high frequency metric

γHF:

γLF =

fc∑i=1

|∆hTD(i)|2, γHF =

N/2+fc−1∑i=N/2

|∆hTD(i)|2, (4.7)

where fc is the cut-off index in the TD, which may be chosen based on the resulting pattern of

TD analysis and may depend on various factors such as the channel and noise model. fc must

be chosen properly, such that high and low frequency metric values for different cases are within

68

exclusive range of values. For analysis, we computed γLF and γHF for 1000 random channel

pairs that correspond to different topologies for LS and LMMSE estimators with fc = 20.

The current channel estimate h1 is computed by an OFDM symbol going through the first

channel, and the OFDM symbol that will be used to compute h2 goes through the first or the

second channel of the pair depending on the four cases. The resultant metrics are presented in

Figs. 4.8 and 4.9. Fig. 4.8 suggests that γLF is clearly distinct for Cases C and D compared to

Cases A and B. The γLF is big, when the channel changes. Moreover, Fig. 4.8 suggests that,

γLF is also bigger for Case B compared to Case A, but not as much as Cases C and D. This is

due to the low frequency components of the impulsive noise. Similarly, Fig. 4.9 suggests that

γHF is big for Cases B and D compared to Cases A and C. For Cases B and D, impulsive noise

is present, and γHF turns out to be bigger compared to Cases A and C. The results for the LS

estimator are similar to LMMSE, where it is also possible to distinguish four cases via γLF and

γHF. In summary, TD analysis suggests that γLF can be considered to find out whether the

channel has changed or not, and γHF can be considered to find out whether impulsive noise is

present or not in addition to the background AWGN.

4.5 Performance Analysis and Proposed Final Scheme

We now propose to exploit the information captured by TD analysis for CE. This is achieved

via pilots spaced widely apart (e.g. Fig. 4.2c) in order to keep the estimation overhead low and

still be able to perform TD analysis. One out of L subcarriers are placed as pilots, and we

choose L = 10. Based on the information from TD analysis, the proposed final scheme switches

between various schemes, unlike the conventional CE schemes that are based on a single scheme.

Different design criteria can be considered, such as minimizing the overhead while keeping the

estimation error below a certain value, or just minimizing the estimation error regardless of

the estimation overhead. In this work, we consider keeping the estimation overhead low and

mitigating the effects of the LPTV channel and the impulsive noise as our design criteria for

69

the proposed algorithm. The estimation schemes we considered in this section3 are listed in

Table 4.2.

Table 4.2. CE Schemes

SCHEME # Description

Scheme A Block-type pilots.

Scheme B Decision directed.

Scheme C Comb-type pilots, and interpolation.

Scheme D Discarding the current estimate based on TD analysis.

Scheme E Comb-type pilots, decisions based on current estimate,recompute estimate based on decisions.

Scheme A is the block-type pilot arrangement, where no data transmission takes place and

all of the subcarriers in the OFDM symbol carry pilots for at least the duration of one AC

cycle of the mains. Therefore, the CE overhead is high. Scheme B is the decision directed CE

approach, where the channel estimate from the previous OFDM symbol is utilized to decode the

current symbol. Then, these decisions are used to update the estimate for the current symbol

and to decode the next symbol. Other than pilot symbols that are used for initialization, this

scheme does not utilize pilots. Hence, the estimation overhead is minimal. Schemes A and B

are studied as a reference for comparison with the remaining schemes that have pilots placed

distantly from each other and TD analysis is performed. Scheme C is the comb-type pilot

arrangement, where one out of L subchannels carry pilot symbols, and the channel estimate

for the whole spectrum is found by interpolation for the non-pilot subcarriers. Scheme D is the

case where the current estimate is discarded and the previous estimate is kept. This scheme is

examined only for Case B, where the channel remains the same and impulsive noise is present.

Scheme E initially computes the channel estimate via comb-type pilots and interpolation. Next,

the current OFDM symbol is detected based on this channel estimate, and decisions are obtained

for non-pilot subcarriers. Then, the initial channel estimate that was obtained by interpolation

3Note that Schemes A–E listed in this section and schemes A–E listed in Section 4.3.2.1 refer to two differentset of schemes.

70

for data-carrying subchannels is replaced by LS estimates as in Equation (4.1), X consisting of

pilot values and the decisions made in non-pilot subcarriers. Scheme E is different from Scheme

B in that it makes use of the current estimate to decode the symbol initially rather than the

estimate from the previous symbol, and it is different from Scheme C in that it updates the

interpolation-based estimate with the decision-based estimate.

We now examine the four cases listed in Section 4.4 along with the five schemes listed in

Table 4.2 in more detail. In order to evaluate the performance of the CE schemes, we com-

puted the normalized minimum mean square error (NMMSE), and the number of subchannels

with decoding errors Ed out of the number of subchannels carrying data Nd. The averages

of NMMSE–in dB–and the decoding errors for 1000 realizations of h1 and h2 are shown in

Table 4.3. For all schemes, LS estimates are employed, and the cut-off frequency is fc = 20 for

TD analysis.

Table 4.3. Performance Analysis

CASE # Scheme A Scheme B Scheme C Scheme D Scheme E

Case A - [NMMSEavg] -47.32 -47.12 -42.04 N/A -42.70

Case A - Ed,avg/Nd N/A 0.61 / 256 1.51 / 229 N/A 1.51 / 229

Case B - [NMMSEavg] -30.96 -31.78 -26.68 -42.04 -29.93

Case B - Ed,avg/Nd N/A 27.97 / 256 43.15 / 229 25.77 / 229 43.15 / 229

Case C - [NMMSEavg] -47.35 -21.03 -42.02 N/A -42.74

Case C - Ed,avg/Nd N/A 192.92 / 256 1.54 / 229 N/A 1.54 / 229

Case D - [NMMSEavg] -30.99 -20.63 -30.10 N/A -29.89

Case D - Ed,avg/Nd N/A 192.53 / 256 43.28 / 229 N/A 43.28 / 229

4.5.1 Case A

Case A is when the channel remains the same (h2 = h1) and the noise is background AWGN.

The amplitude and phase of the channel estimates for a single realization of the channel and

noise are depicted in Fig. 4.10. We note that the magnitude of the channel estimates for Schemes

71

Figure 4.10. Case A channel estimates for the various schemes. The amplitudeand phase of the channel transfer function are illustrated.

A and B are the same for all cases due to the constant envelope QAM symbols, but the phase

is different when the symbol is detected in error for a subchannel for Scheme B. Among all

schemes, Scheme A has the minimum NMMSE of -47.32 dB, and Scheme B performs very close

to Scheme A with -47.12 dB NMMSE. Scheme B, the decision directed approach, has a valid

estimate from the previous OFDM symbol to decode the symbol since the channel remained

the same for Case A. Hence the estimate is a good one, and the average number of decoding

errors is as low as 0.61 out of 256 subchannels. On the other hand, Scheme C has about 10%

overhead4 due to the scarcely placed pilots for TD processing: 27 subchannels carry pilots, and

4The overhead for Schemes C, D and E is the same for all cases, since they are based on TD processing madeavailable by comb-type pilot arrangement with L = 10. Since the first and the last subchannels are also assignedpilots in the comb-type arrangement, there are a total of 27 subchannels out of 256, which corresponds to the

72

229 subchannels carry data. The NMMSE is -42.04 dB, and the average number of decoding

errors is 1.51 out of 229 data-carrying subchannels. The NMMSE is about 5 dB more than

Schemes A and B, but this does not affect the decoding of the data carrying subchannels, since

the decoding error is small. As seen in Fig. 4.10, the estimates are very close to each other even

for subchannels with high signal attenuation. Hence, when these decisions for data carrying

subchannels and the pilot subchannels are used to recompute the LS estimate in the case of

Scheme E, the NMMSE is improved about 0.7 dB compared to Scheme C. For Scheme E, the

estimation overhead is reduced to about 10% from the 100% overhead of Scheme A, and a

comparable CE with low NMSSE is achieved. Compared to Scheme B, Scheme E provides a

comparable CE performance with a comparable raw data rate. Data can be carried in 229

subchannels as opposed to 256 subchannels. However, Scheme B lacks TD analysis, which

results in performance degradation for the remaining Cases B, C, and D. Therefore, for Case

A, we propose to employ Scheme E due to its low overhead and TD analysis.

4.5.2 Case B

In Case B, the channel remains the same (h2 = h1) and impulsive noise is present in the

channel along with background AWGN. The amplitude and phase of the channel estimates for

a single realization of the channel and noise are depicted in Fig. 4.11. In this case, the channel

estimates for Schemes A, B, C, and E are affected badly by the presence of impulsive noise in

the channel. The NMMSE values for these schemes are -30.96, -31.78, -26.68, and -29.93 dB,

respectively. This is a result of the received noisy symbol with impulsive noise. Hence, for this

case, we propose to exploit the information from TD analysis that the channel has not changed

and employ Scheme D. Scheme D is simply aware of the presence of impulsive noise via TD

processing, and discards the channel estimate that is computed using the current noisy symbol.

It instead utilizes the previous estimate. The NMMSE for Scheme D is -42.04 dB, which is much

smaller than all other schemes. In terms of raw data rates, Scheme A has no data-carrying

subchannels, so the raw data rate is zero. Scheme B has an average of 27.97 decoding errors

actual 10.54% overhead.

73

Figure 4.11. Case B channel estimates for the various schemes. The amplitudeand phase of the channel transfer function are illustrated.

out of 256 subchannels, and Scheme D has a comparable average number of errors, 25.77 out of

229 compared to Scheme B. Additionally, Scheme D is aware of the presence of impulsive noise

and the resultant high number of decoding errors via TD processing. Compared to Schemes

C and E, Scheme D has a smaller decoding error average: 25.77 compared to 43.15. This is

because Scheme D employs a better channel estimate than Schemes C and E to decode the

noisy symbol. Therefore, the errors are only due to the increased noise level, but not due to the

bad CE as for Schemes C and E. Hence, we propose to utilize Scheme D for Case B due to its

better CE compared to all other schemes, low overhead compared to Scheme A, and relatively

low decoding errors compared to Schemes C and E.

74

4.5.3 Case C

Figure 4.12. Case C channel estimates for the various schemes. The amplitudeand phase of the channel transfer function are illustrated.

Case C is when the channel changes (h2 6= h1) and the noise is background AWGN. The

amplitude and phase angle of the channel estimates for a single realization of the channel and

noise are depicted in Fig. 4.12. In this case, Scheme A has the minimum NMMSE of -47.35

dB among all schemes. Since Scheme B uses the previous channel estimate to decode the

symbols, and the channel has changed, this results in a large number of decoding errors and

a bad estimate for the channel. The average number of decoding errors in this case is 192.92

out of 256 subchannels, and the NMMSE is -21.03 dB. Since QAM is used for modulation, the

magnitude responses for Schemes A and B are the same, but the phase responses for Schemes A

75

and B are very different from one another. The reason for this is the large number of decoding

errors for Scheme B. Scheme B in this case suffers the most among all schemes from the change

in the channel transfer function. It has the worst channel estimate and the highest number

of decoding errors among the data-carrying schemes B, C, D and E. For Schemes C and E,

the average number of decoding errors is 1.54 out of 229 data-carrying subchannels, and the

NMMSE is -42.02 and -42.74 dB, respectively. Both schemes utilize the current estimate—

computed using the pilots placed widely apart and interpolation—to decode the symbols. This

results in smaller number of decoding errors compared to Scheme B. This is the reason for 0.7

dB improvement in the channel estimate when these decisions are considered to replace the

estimates obtained by interpolation for the non-pilot subcarriers. For this reason we propose to

utilize Scheme E in this case, which has small number of decoding errors, a comparable channel

estimate, and a much lower overhead for CE compared to Scheme A.

4.5.4 Case D

This is the case where the channel changes (h2 6= h1) and impulsive noise is present. The

amplitude and phase angle of the channel estimates for a single realization of the channel and

noise are depicted in Fig. 4.13. In this case, Schemes A, C and E have similar CE performance

with NMMSE of -30.99, -30.10, and -29.89 dB, respectively. Scheme A results in noisy estimates

due to the presence of impulsive noise similar to Case B. Scheme B utilizes the previous channel

estimate, and provides a poor estimate and high decoding error, since the channel has changed

and the impulsive noise is present. The NMMSE for Scheme B is -20.63, the worst among all

schemes. The average number of decoding errors for Scheme B is 192.53 out of 256, which is

much higher than Schemes C and E. The average number of decoding errors for Schemes C and

E is 43.28, similar to Case B, but higher than Cases A and C due to the presence of impulsive

noise. In this case, we propose to utilize Scheme C, because of its comparable CE and lower

overhead compared to Scheme A, and slightly better CE than Scheme E. Scheme E does not

improve the CE for Scheme C in this case, because of relatively high decoding errors compared

76

Figure 4.13. Case D channel estimates for the various schemes. The amplitudeand phase of the channel transfer function are illustrated.

to Cases A and C.

4.5.5 Proposed Final Scheme

We illustrate the proposed final CE scheme based on the TD analysis in Fig. 4.14 to sum-

marize our findings in this section. In the figure, the delay block D is a delay of one OFDM

symbol time. When a new OFDM symbol is received, TD analysis is performed using the

channel estimate from the previous symbol and an initial estimate of the current symbol based

on comb-type arranged pilots and interpolation for the current symbol. HF and LF metrics

are computed and the case type is determined based on whether the channel has changed or

the impulsive noise is present. Then, depending on the case, the corresponding CE Scheme

77

Channel estimate from previous symbol

TD AnalysisHF and LF metriccomputation

CurrentOFDM Symbol

Scheme E

Scheme D

Scheme E

Scheme C

Case A

Case B

Case C

Case D

Currentchannel estimate

D

Figure 4.14. Proposed final channel estimation scheme based on TD analysis.

is utilized and the current channel estimate is computed. The advantage of performing TD

analysis is the ability to utilize various type of CE schemes depending on the case, rather than

using one conventional scheme for all cases. Based on how frequent the channel changes, and

how frequent impulsive noise occurs in the power line, the amount of improvement in estimation

accuracy compared to block-type pilot based and decision directed approaches will be different.

The proposed final scheme is robust to changes in the channel and the noise compared to block-

type pilot arrangement and decision directed approaches, and provides a low overhead solution

for the BB PLC channel.

4.5.6 Optimal Schemes

Similar to the comb-type pilot arrangement, TD analysis can be combined with other CE

schemes such as the block-type pilot arrangement and the decision directed scheme. If the

design criteria is to minimize the NMMSE regardless of the estimation overhead, the optimal

scheme would be the block-type pilot arrangement combined with TD analysis. Analyzing the

performance results in Table 4.3, a final scheme that utilizes Scheme A for Cases A-C-D, and

Scheme D for Case B would be optimal. On the other hand, if the design criteria is to minimize

the estimation overhead regardless of NMMSE, the decision directed approach Scheme B would

78

be optimal. In this case, TD analysis can be performed on the decision directed estimates for

two consecutive OFDM symbols in some cases to improve the performance of Scheme B. For

instance for Cases A and B, considering the performance results in Table 4.3, NMMSE for

Scheme B is about 5 dB less than Scheme C. Since TD analysis works well using the estimates

of Scheme C, it also works well using the estimates of Scheme B for Cases A and B. Therefore,

TD analysis for Case B results in dependable metrics and can be combined with Scheme B

for Case B, provided that the previous channel estimate is one with a low NMMSE that has

converged over time. Then, an improved version of Scheme B in terms of smaller NMMSE

would utilize Scheme D instead of Scheme B for Case B.

4.6 Summary and Conclusions

In this chapter, we first examined LS and LMMSE estimators for pilot-based CE with

different pilot geometry. We implemented incline-type pilot arrangement for BB PLC channels

in order to help reduce interpolation error inherent in comb-type pilot arrangement. When

incline-type pilot arrangement is utilized, the estimates for each subchannel is a combination of

pilot based and channel based estimates, unlike the comb-type, where the estimates for non-pilot

subchannels is only based on interpolation.

We then analyzed a new method for CE for the BB PLC channel based on TD analysis and

pilots spaced widely apart. TD analysis is employed in identifying abrupt changes in the power

line channel and the noise by examining the change in the channel estimate for two consecutive

OFDM symbols. We defined four cases depending on the changes in the channel and the noise.

For each case, five CE schemes are studied, and the proposed final scheme utilizes the scheme

with the low or the lowest estimation error and low overhead. The block type pilot based

CE Scheme A has the highest overhead, and the decision directed Scheme B suffers when the

channel changes and impulsive noise is present with poor CE and high decoding errors. Scheme

E utilizes decisions for non-pilot subcarriers unlike Scheme C, which performs interpolation for

these subcarriers. For cases where the decoding error is small and the decisions are reliable,

79

Scheme E outperforms Scheme C, since there is no interpolation error introduced and decisions

are mostly accurate. If the channel does not change and impulsive noise is present, it is better to

discard the estimate and keep the previous estimate, which is Scheme D. This can be achieved

by performing TD analysis. Hence, the proposed final scheme is aware of the changes in the

channel and the noise via TD analysis, and switches between various schemes based on their

performance with low estimation error and overhead, which makes it a good candidate for the

BB PLC channel.

Additionally, the idea of TD analysis can also be combined with block-type pilot arrange-

ment and decision directed CE schemes. The optimal scheme that minimizes NMMSE can be

achieved by employing TD analysis with block-type pilot arrangement, and the optimal decision

directed CE scheme that minimizes CE overhead can be improved by employing TD analysis

in terms of estimation accuracy for certain cases.

80

Chapter 5

Conclusion

The objective of this work is to develop bit loading and CE schemes specifically designed for

BB PLC channels. PLC has been a topic of interest for the past decades due to its economic

advantage that it does not require additional investment for infrastructure, and is one of com-

munication technologies envisaged for SG. However, since the power lines are not designed for

data communication, challenges are involved in designing a communication system for power

lines. In particular for BB PLC, LPTV channel and the impulsive noise are two characteristics

of power lines that have to be taken into consideration. The main contributions of our work

are: (i) LPTV-aware bit loading (ii) CE based on TD analysis, for an OFDM system. These

contributions are listed below in more detail.

5.1 Contributions

1. Simplistic adaptation schemes in the literature lack the ability to view the entire AC mains

cycle when allocating bits and power in each microslot. Therefore, they are not optimal

for LPTV channels. Optimal LPTV-aware bit loading scheme based on a greedy-type

algorithm is developed.

2. By utilizing representative values from microslots, the complexity of the optimal algorithm

is greatly reduced without sacrificing performance. Power clipping is utilized to avoid peak

81

power levels for the reduced complexity algorithm.

3. The idea of LPTV-aware bit loading, complexity reduction, and power clipping is im-

plemented for even-like power distribution algorithms and similar gains in bit rates are

achieved.

4. LS, LMMSE algorithms with linear and cubic interpolation are implemented and com-

pared from a practical perspective for BB PLC channels.

5. Incline-type pilots are examined to help reduce interpolation error inherent in comb-type

pilot arrangement, and a comparison of block-type, comb-type, and incline-type pilot

arrangements is made.

6. TD analysis for CE is developed. By examining the frequency content of the difference

between two channel estimates, crucial information such as the cause for an abrupt change

in the channel estimate can be extracted. TD analysis determines the changes due to a

change in the transfer function and noise by computing high frequency and low frequency

metrics in the TD.

7. A robust scheme that utilizes pilots placed widely apart and TD analysis is developed.

The advantage of exploiting information from TD analysis is the ability to switch between

various CE schemes based on the knowledge of a change in the transfer function or the

noise. Unlike conventional CE schemes based on a single approach, CE scheme based

on TD analysis is aware of the abrupt changes in the channel whether due to the LPTV

channel or the impulsive noise.

5.2 Areas of Further Study

The bit loading and CE problems are treated separately in this work and solutions are

provided for each problem. The interdependency of these two problems—for instance the effect

of CE in error on the performance of bit loading—can be evaluated by investigating a system

that utilizes both LPTV-aware bit loading and CE.

82

New TD analysis metrics can be developed for different power line noise models in the

literature, and their performance can be evaluated. In this work, we developed TD analysis

based on pilot symbols; however, the application of TD analysis to various decision directed

schemes and their performances is an interesting topic to investigate in the future.

83


84

Appendix A

Matroid Structures

A matroid M=(S, I) is an algebraic structure composed of the following [19]:

• S = {s1, s2, s3, . . .}, a finite set of elements, and

• I = {S0,S1,S2, . . .}, a collection of subsets of S,

where I satisfies the following conditions:

(C.1) The empty set ∅ is a member of I.

(C.2) If A ⊆ Si and Si ∈ I, then A ∈ I; a subset of a subset in I is also included in I.

(C.3) If Si ∈ I and Sj ∈ I, and |Si| < |Sj |, then there exists an element s ∈ Sj −Si such

that Si ∪ {s} ∈ I, where |S| is the cardinality of set S.

S is the ground set of a matroid, and the members of I are independent sets of a matroid.

The rank is the cardinality of the largest member of I, and S∗ ∈ I is maximal when it is not a

subset of any other member of I.

A simple example is the uniform matroid of rank c over the ground set S = {0, 1, 2, . . . L−1}

with L elements and the collection of subsets of S, Ic = {S ⊆ S : |S| ≤ c}, for which the

cardinality does not exceed c. Since (S, Ic) satisfies all three conditions listed above, it is a

matroid with rank c.

85

Algorithm 2 Greedy algorithm to solve the optimization problem A.2

1: while an element s ∈ S −S exists such that S ∪ {s} ∈ I do2: Select an element s in S −S such that: (i) S ∪ {s} ∈ I, and (ii) w(s) is minimum overS −S

3: Update S = S ∪ {s};4: end while5: exit.

A.1 Matroids and Greedy Algorithms

Matroids can be used for solving resource allocation problems related to greedy algorithms.

Greedy algorithms are iterative searching algorithms that always move in the direction of largest

increment (decrement) to the function to be maximized (minimized), and they only proceed in

the forward direction and do not trace back.

In order to introduce greedy algorithms, the following are defined. Let the independence

system over a finite set S be (S, I), where I={S0,S1, . . .} is a collection of subsets of S, which

only satisfies conditions C.1 and C.2. Let us assume that each element in S has a weight that

can be described by a weight function w : S → R, and the weight W (S) of a subset S of S is

the sum of the weight of its elements

W (S)=∑s∈S

w(s), and W (∅) = 0. (A.1)

The optimization problem to solve for the independence system matroid is to find a maximal

member S of I with minumum weight W (S):

find a maximal S ∈ I with minumum W (S). (A.2)

The pseudocode for the greedy algorithm for this optimization problem is given in Algorithm 2.

The algorithm starts with an empty set for S, and builds S iteratively at each step by adding

the minimum weight element s of S −S, such that the updated S is also included in I. The

algorithm stops when last condition can no longer be met, and final set S is guaranteed to be

86

maximal. This is based on the following property:

Property 1: (Edmonds [29]) For any assigned weight function w : S → R, the greedy

algorithm solves the problem A.2 if and only if (S, I) is a matroid.

A.2 Integer Bit Loading as a Matroid Structure

Consider an OFDM system with N subchannels. Let us assume that the total number of

bits allocated to N subchannels is fixed to Btot = c. The pair (k, i) represents the k-th bit

transmitted in the i-th subchannel. For each subchannel, assume that there exists a maximum

number of bits that can be transmitted at that subchannel (i.e. for i-th sunchannel bmaxi ).

Then, the collection of all of these pairs forms the finite set:

S={(k, i) : 0 ≤ k ≤ bmaxi , 0 ≤ i ≤ N − 1}. (A.3)

For the ground set S and assumed cardinality c = Btot, the collection of all possible subsets of

S with cardinality not exceeding c is given by

Ic = {S ⊆ S : |S| ≤ c}. (A.4)

In (A.4), Ic includes all possible bit loading patterns, for which Btot = c, and (S,Ic) in (A.3)

and (A.4) is a uniform matroid of rank c = Btot. This proves the matroid structure of the

following integer bit loading maximization for any value of c:

max{bi,i=0,...,N−1}

Btot =

N−1∑i=0

bi, (A.5)

subject to three constraints:N−1∑i=0

εi ≤ εtot, (A.6a)

bi ∈ {0, 1, 2, . . . , bmaxi }, i = 0, 1, 2, . . . , N. (A.6b)

87

Hence, the application of greedy Algorithm 2 with the weight function being the excess energy

required to transmit k-th bit in the i-th subchannel pair in S:

w(k, i) = ∆εi(k), 0 ≤ k ≤ bmaxi , i = 0, 1, 2, . . . , N − 1, (A.7)

results in finding the maximal member S∗ of Ic with cardinality c = Btot = |S∗| and minimum

weight

W (S∗) =∑

(k,i)∈S∗

∆εi(k). (A.8)

The rate function for subchannel i can be written as

R(σi) = log2

(1 +

σiΓ

)= log2

(1 + εi

|Hi|2

N0(i)Γ

)(bits/symbol),

where σi is the overall SNR at subchannel i, Γ is the SNR-gap which accounts for non-ideal

issues in implementation, Hi is channel frequency response at subchannel i, and N0(i) is the

noise power spectral density. The rate function for the ideal case of no SNR-gap as a function of

σi = εi(|H(i)|2/N0(i)) is illustrated in Fig. A.1. When this relation applies, the energy required

to transmit k bits at subchannel i is

εi(k) =ΓN0

|Hi|2(2k − 1), (A.9)

where the expression outside the paranthesis is the same for different k for subchannel i. There-

fore,

∆εi(k) = 2∆εi(k − 1). (A.10)

This implies that the greedy algorithm picks pairs corresponding to a subchannel starting from

the first bit in an increasing fashion, since it requires more energy to transmit the next bit in

a subcahnnel compared to the previous bits transmitted, which results in a bit loading pattern

that practically makes sense. Due to the strictly increasing concave nature of the rate function

88

Figure A.1. Strictly increasing concave rate function illustrated as a function ofoverall SNR with ideal case of no SNR-gap.

for a communication channel, the greedy algorithm solution converges to the globally optimal

bit allocation for the maximization problem. From an implementation point of view, Btot is

unknown at the beginning of the bit loading process. However, the greedy algorithm can be

applied in an iterative fashion starting at an initial Btot = 0, then increasing Btot by one at

each step.

89


90

Appendix B

Channel Interpolation

If the number of subchannels carrying pilot symbols is Np and the total number of sub-

channels is N , the set of Np complex values is interpolated to obtain the channel frequency

response. The interpolation is a linear filtering process that can be performed in frequency

or time domain. When one out of L subchannels are placed pilot symbols, for the comb-type

geometry, the frequency index of pilot subcarriers is k = nL, where L is the pilot spacing, and

0 ≤ n ≤ Np − 1. The CE algorithms (i.e. LS, LMMSE etc.) estimate the channel at pilot

locations, then interpolate to estimate the remaining locations. The received symbol at the

pilot locations can be expressed as

Yp = XpHp + np, (B.1)

where Xp is the complex pilot QAM symbol transmitted at the pilot location, Hp is the cor-

responding channel response, np is the complex additive noise, and p = nL. The LS channel

estimate is then

Hp = Yp/Xp = Hp + ep, (B.2)

where ep is the estimation error given by

ep = np/Xp. (B.3)

91

Figure B.1. Linear and cubic interpolation kernels.

For the i-th subchannel, where 0 ≤ i ≤ N − 1, the channel estimate via interpolation is defined

by the following relation

Hi =∑n

HnL.u

(i− nLL

), (B.4)

where u(x) is the interpolation kernel [22] (i.e. linear, cubic, nearest neighbor, B-spline, DFT

zero padding, second order Gaussian etc.). For instance, u(x) for linear interpolation is given

by

u(x) =

1− |x| 0 ≤ |x| ≤ 1

0 elsewhere,(B.5)

and for cubic interpolation

u(x) =

32 |x|

3 − 52 |x|

2 + 1 0 ≤ |x| ≤ 1

−12 |x|

3 + 52 |x|

2 − 4|x|+ 2 1 ≤ |x| ≤ 2

0 elsewhere.

(B.6)

The linear and cubic interpolation kernels are illustrated in Fig. B.1.

92

Frequency indexn0L (n0+1)L (n0+2)L

H n0L

n0L

i0,int

i0

i0,noise

HiHie

e

e

Figure B.2. Estimation error decomposition [22, Fig. 3]

.

B.1 Estimation Error

The estimation error ei for the i-th subchannel is the difference between the true channel

response at that subchannel and the estimated value:

ei = Hi − Hi

=

(Hi −

∑n

HnL.u

(i− nLL

))−

(∑n

enL.u

(i− nLL

)). (B.7)

Equation B.7 is composed of two error sources: (i) interpolation error, and (ii) noise error:

ei = ei,int + ei,noise, (B.8)

where

ei,int = Hi −∑n

HnL.u

(i− nLL

), (B.9)

93

ei,noise =∑n

enL.u

(i− nLL

). (B.10)

If we assume that there is no noise in the communication channel, this results in perfect channel

estimates with no error at the pilot locations. However, the non-pilot subchannels’ estimates

will be obtained by interpolation, and the estimation error for the non-pilot subchannels will

only be caused by interpolation in this case. This error is the interpolation error portion of the

actual estimation error. The interpolation error depends on the interpolation kernel, channel

characteristics, and pilot spacing L. The noise error is the second component of estimation error

and is a function of noise at pilot subcarriers and the interpolation kernel. The decomposition

of estimation error in terms of interpolation and noise errors is illustrated in Fig. B.2 for real

Hi and linear interpolation.

94

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